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In a medium saucepan, melt the butter over medium heat.
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Add the dark brown sugar and stir until it dissolves and begins to bubble, about 2-3 minutes.
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In a small bowl, whisk together the heavy cream, cornstarch, and salt until smooth.
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Slowly add the cream mixture to the saucepan, whisking constantly to prevent lumps.
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Continue to cook, whisking frequently, until the mixture thickens, about 5-7 minutes.
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Remove the pan from the heat and stir in the vanilla extract.
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In a separate bowl, lightly beat the egg yolks. Slowly add a small amount of the hot butterscotch mixture to the yolks, whisking constantly to temper them.
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Gradually add the tempered egg yolks back into the saucepan, whisking continuously.
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Return the saucepan to medium heat and cook for an additional 2-3 minutes, until the filling is thick and smooth.
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Pour the butterscotch filling into the cooled crust and spread it evenly.
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Chill the tart in the refrigerator for at least 4 hours or until set.