Beat the softened butter with both sugars on medium speed for 3 minutes until pale and fluffy. Add 2 eggs one at a time, then mix in 2 tsp vanilla.

Whisk together flour, baking soda, and 1/2 tsp salt in a separate bowl. Stir the dry mixture into the butter mixture on low speed until just combined.

Line 2 baking sheets with parchment. Scoop the dough into 12 balls of about 3 tbsp each, press into disks 3/4 inch thick, and press a shallow well into the center of each.

In a small saucepan over medium heat, stir together corn syrup, 1/4 cup brown sugar, and 2 tbsp butter until the butter melts and the mixture simmers, about 3 minutes. Remove from heat, cool 5 minutes, then whisk in the remaining egg, 1/2 tsp vanilla, and 1/4 tsp salt. Stir in pecans.

Spoon about 2 tbsp of filling into each cookie well. Bake one sheet at a time at 350°F (175°C) on the middle rack for 13 to 15 minutes, until edges are set and lightly golden.

Cool on the pan for 10 minutes, transfer to a rack, then arrange on a serving board and press a whole pecan into each center with a pinch of flaky sea salt on top.