In a small saucepan over medium heat, melt the butter. Continue cooking, swirling the pan constantly. It will foam up, then start to crackle. Once you see golden-brown bits at the bottom and it smells like toasted hazelnuts, remove from heat immediately. Pour into a large heat-proof bowl and let it cool for about 15 minutes.
To the cooled brown butter, add both sugars. Whisk vigorously until the mixture is smooth and no longer looks oily.
Add the egg, egg yolk, and vanilla. Whisk for about 1–2 minutes until the batter lightens in color and becomes slightly thickened.
Fold in the flour, baking soda, and salt using a spatula until just a few streaks of flour remain.
Gently stir in the caramel bits and chocolate chips.
Scoop large rounds of dough (about 3 tablespoons each) onto a tray. Cover and chill in the fridge for at least 2 hours, or overnight.
Preheat your oven to 350°F (175°C). Place chilled dough balls on a parchment-lined baking sheet, spaced 3 inches apart. Bake for 10–12 minutes until the edges are golden but the centers still look slightly soft.
While warm, sprinkle generously with flaky sea salt. Let them cool on the pan for 10 minutes to allow the caramel to set.
