In a medium skillet over medium heat, melt 1 tablespoon butter. Add chopped pecans and toast for 4 to 5 minutes, stirring constantly until fragrant and darkened. Transfer to a bowl to cool.

In the same skillet, melt 1 cup (2 sticks) unsalted butter over medium heat. Swirl occasionally as it foams and sizzles. After 5 to 7 minutes, nutty golden-brown specks will settle at the bottom and a heavenly toffee aroma will rise. Immediately pour into a heatproof bowl and let cool for 15 minutes.

In a medium bowl, whisk together flour, baking soda, and fine salt.

In a large bowl, whisk the cooled brown butter with brown sugar and granulated sugar until well combined. Whisk in the whole egg, egg yolk, and vanilla extract until glossy and thick (about 1 minute).

Fold in dry ingredients with a rubber spatula just until no flour pockets remain. Fold in the toasted chopped pecans. Chill dough for 30 minutes in the refrigerator.

Preheat oven to 350°F (175°C). Line two large baking sheets with parchment paper.

Scoop 2-tablespoon balls of chilled dough onto baking sheets, spaced 2 inches apart.

Bake for 10 to 12 minutes until edges are set and golden brown, but centers are still soft and slightly puffed.

Immediately sprinkle with flaky sea salt. Allow cookies to cool on the baking sheet for 5 minutes, then transfer to a wire cooling rack.