In a large mixing bowl, combine all-purpose flour and fine sea salt. Cut in the cold unsalted butter using a pastry cutter or forks until the mixture resembles coarse crumbs. Gradually add ice-cold water, one tablespoon at a time, mixing until the dough holds together. Shape the dough into a disk, wrap it in plastic wrap, and refrigerate for 30 minutes to ensure a flaky crust.

Once chilled, remove the dough from the refrigerator and roll it out on a lightly floured surface to about 1/8 inch thickness. Use a round cutter to cut out circles that fit your muffin tin. Gently press the dough rounds into the muffin tin to form little pastry cups, ensuring there are no air pockets. Preheat your oven to 375°F while preparing the filling.

In a medium bowl, whisk together the packed brown sugar, melted unsalted butter, large egg, and vanilla extract until smooth. Stir in the heavy cream, corn syrup, and a pinch of salt. Ensure every ingredient is well blended, achieving a creamy consistency that's free of lumps.

Spoon the luscious filling into each pastry-lined muffin cup, filling them about two-thirds full to avoid overflow during baking.

Carefully place the muffin tin in the preheated oven and bake for 12-15 minutes. The edges should be set and golden brown, while the centers will still have a slight jiggle.

After baking, allow the butter tarts to cool in the pan for 10 minutes. Once cooled, gently transfer the tarts to a wire rack to cool completely.