Preheat your oven to 400°F. Grease a 12-cup muffin tin or line it with baking cups.
In a saucepan over medium heat, melt ¼ cup (4 tablespoons) of unsalted butter. Stir in the brown sugar, corn syrup, heavy cream, egg, vanilla extract, walnuts, and salt. Cook for a few minutes, stirring occasionally, until the mixture thickens slightly. Remove from heat and let cool slightly.
Roll out the thawed puff pastry sheets and cut them into 6 equal rectangles. Gently press each piece into the muffin tin, creating small cups.
Spoon about ¼ cup of the walnut filling into each puff pastry cup and brush the edges with the whisked egg.
Bake in the preheated oven for 15-20 minutes, or until the filling is bubbly and the puff pastry is golden brown. Monitor closely to prevent over-browning.
Let the tarts cool in the muffin tin for about 5 minutes, then carefully transfer them to a wire rack to finish cooling. Serve warm or at room temperature.
