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Whisk together milk and cornstarch in a medium bowl.
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Melt the butter in a large heavy saucepan over medium heat. Add the brown sugar and cook, frequently whisking, for 2 - 3 minutes until the mix is smooth and bubbling. Slowly and carefully pour in the milk mix - careful it may splatter a bit - whisking constantly. Cook while whisking for about 2 minutes until the mix is hot and there are no lumps of sugar left. Remove from heat.
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Whisk together the egg yolks and salt in a medium bowl. Slowly pour in the milk mix whisking constantly. Return the mix to the saucepan and cook over medium low heat, for 6 - 8 minutes always whisking. Cook mix until it comes to a boil and large bubbles rise to surface.
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Immediately pour the mix through a fine strainer set over a bowl. Whisk in the lemon juice and vanilla. Let cool for 15 minutes, occasionally whisking to keep a skin from forming.
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Divide pudding among 4 ramekins or custard cups. Place a piece of plastic wrap on the surface of pudding (directly on the pudding) so no skin forms. Let cool to room temperature and then refrigerate for 3 hours or til thoroughly chilled and set. You can also refrigerate upto 2 days if making ahead.