Place racks in upper and lower thirds of oven; preheat to 375°. Whisk flour, kosher salt, and baking soda in a small bowl; set aside.

Cook ½ cup (1 stick) butter in a large saucepan over medium heat, swirling often and scraping bottom of pan with a heatproof rubber spatula, until butter foams, then browns, about 4 minutes. Scrape melted butter and any brown bits to a large heatproof mixing bowl and let cool 1 minute. Cut remaining ¼ cup (½ stick) butter into small pieces and add to brown butter (it should start to melt but not foam and sizzle, so test with one piece before adding the rest).

Once butter is melted, add dark brown sugar and granulated sugar and whisk, breaking up any clumps, until sugar is incorporated and no lumps remain. Add egg and egg yolks and whisk until sugar dissolves and mixture is smooth, about 30 seconds. Whisk in vanilla extract.

Using rubber spatula, fold reserved dry ingredients into butter mixture just until no dry spots remain, then fold in chocolate chips (the cookie dough will be soft but should hold its shape once scooped; if it slumps or oozes after being scooped, stir dough back together several times and let rest 5–10 minutes until scoops hold their shape as the flour hydrates).

Using a 1½-ounce scoop (3 tablespoons), portion out 16 balls of dough and divide between 2 parchment-paper-lined rimmed baking sheets. Bake cookies, rotating sheets if cookies are browning very unevenly (otherwise, just leave them alone), until deep golden brown and firm around the edges, 8–10 minutes. Let cool on baking sheets.