A collection of sources telling a similar story:
H.A. Ironside, In the Heavenlies (Ephesians), 1935:
It has been pictured in this way. Here is a vast host of people hurrying down the broad road with their minds fixed upon their sins, and one stands calling attention to yonder door, the entrance into the narrow way that leads to life eternal. On it is plainly depicted the text, “Whosoever will, let him come.” Every man is invited, no one need hesitate. Some may say, “Well, I may not be of the elect and so it would be useless for me to endeavor to come, for the door will not open for me.” But God’s invitation is absolutely sincere, it is addressed to every man. “Whosoever will, let him take of the water of life freely” (Revelation 22:17). If men refuse to come, if they pursue their own careless, godless way down to the pit, whom can they blame but themselves for their eternal judgment? The messenger addressed himself to all, the call came to all, the door could be entered by all but many refused to come and perished in their sins. Such men can never blame God for their eternal destruction. The door was open, the invitation was given, they refused and He says to them sorrowfully, “Ye will not come unto me that ye might have life.” But see, as the invitation goes forth every minute or two some one stops and says, “What is that?” “The way to life,” is the reply. “Ah, that I might find the way to life! I have found no satisfaction in this poor world.” We read, “She that liveth in pleasure is dead while she liveth.” “I should like to know how to be free from my sin, how to be made fit for the presence of God.” And such an one draws near and listens and the Spirit of God impresses the message upon his heart and conscience and he says, “I am going inside: I will accept the invitation; I will enter that door,” and he presses his way in and it shuts behind him. As he turns about he finds written on the inside of the door the words, “Chosen in Christ before the foundation of the world.” “What!” he says, “had God His heart fixed on me before ever the world came into being?” Yes, but he could not find it out until he got inside. You see, you can pass the door if you will, you can trample the love of God beneath your feet, you can spurn His grace if you are determined to do it, but you will go down to the pit and you will be responsible for your own doom.
C.H. Spurgeon, A Sermon for the most miserable of men’ NO. 853 January 31, 1869:
I see indelible marks both of predestination and free agency everywhere in God’s universe. Then why do you ask questions about your election when God says, “whosoever will”? It is foolish to stand and ask whether you are ordained to come when the invitation bids you come. Come, and you are ordained to come; stay away, and you deserve to perish. Yonder is the gate of the hospital for sick souls, and over it is written, “Whosoever will, let him come,” and you stand outside that house of mercy, and say, “I do not know whether I am ordained to enter.” There is the invitation, man! Why are you so mad? Would you talk like that at Guy’s or at Bartholomew’s Hospital? Would you say to the kind persons who picked you up in the street, and carried you to the hospital, “Oh, for goodness’ sake, do not take me in, I do not know whether I am ordained to go in or not”? You know the hospital was built for such as are sick and wounded, and when you are taken in you perceive that it was built for you. I do not know how you are to find whether you were ordained to enter the hospital or not, except by getting in; and I do not know how you are to find out your election to salvation, except by trusting Jesus Christ, who bids you trust, and promises that if you do so you shall be saved.
John Bunyan, Pilgrim’s Progress:
So, in process of time, Christian got up to the gate. Now, over the gate there was written, Knock, and it shall be opened unto you. Matt 7:8
James Montgomery Boice quoting Donald Grey Barnhouse in The Gospel of John: An Expositional Commentary (Baker, Grand Rapids, MI 2005), 744:
A number of years ago a woman sat in a pew in the Tenth Presbyterian Church in Philadelphia, which I now serve as pastor. At the time, the pastor was Donald Grey Barnhouse. He was talking about the cross and of the need to believe on the Christ who died upon it. The woman I am talking about was not a Christian. She had been raised in a religious home and had heard about Jesus. She had heard about the cross. But she did not understand these things and therefore obviously had never actually trusted in Jesus for her salvation. In order to make clear that for salvation only belief in Jesus Christ is necessary Barnhouse said, ‘Imagine that the cross has a door in it. All you are asked to do is to go through. On one side, the side facing you, there is written an invitation: “Whosoever will, may come.” You stand there with your sin upon you and wonder if you should enter or not. Finally you do, and as you do the burden of your sin drops away. You are safe and free. Joyfully you then turn around and see written on the backside of the cross, through which you have now entered, the words “Chosen in him before the foundation of the world.”’ Barnhouse then invited those who were listening to enter. The woman later said that this was the first time in her life that she had really understood what it meant to be a Christian and that in understanding it, she had believed. She believed right there—in that church at that moment. She entered the door. Moreover, the rest of her life bore witness to the fact that a great change had occurred and that she was God’s child.” Boice concludes the story with these words, “I am certain of the facts of this story because that woman was my mother.”
Source: https://www.puritanboard.com/threads/calvinism-sign-outside-and-inside-building.90878/, Comments 3 and 4