Cook butter in a medium saucepan over medium heat, stirring often, until it foams, then browns, 5–8 minutes. Scrape into a large bowl and let cool slightly., about 10 minutes (an instant-read thermometer should register 125°).

Meanwhile, whisk flour, baking soda, and kosher salt in a medium bowl.

Add brown sugar and granulated sugar to browned butter. Using an electric mixer on medium speed, beat until incorporated, about 1 minute. Add eggs and vanilla, increase mixer speed to medium-high, and beat until mixture lightens and begins to thicken, about 1 minute. Reduce mixer speed to low; add dry ingredients and beat just to combine. Mix in toffee pieces and chocolate wafers with a wooden spoon or rubber spatula. Let dough sit at room temperature at least 30 minutes to allow flour to hydrate. Dough will look very loose at first, but will thicken as it sits.

Place a rack in middle of oven; preheat to 375°. Using a 1-oz. ice cream scoop, portion out balls of dough and place on a parchment-lined baking sheet, spacing about 3" apart (you can also form dough into ping pong–sized balls with your hands). Do not flatten; cookies will spread as they bake. Sprinkle with sea salt.

Bake cookies until edges are golden brown and firm but centers are still soft, 9–11 minutes. Let cool on baking sheets 10 minutes, then transfer to a wire rack and let cool completely. Repeat with remaining dough and a fresh parchment-lined cooled baking sheet.

Do Ahead: Cookie dough can be made 3 days ahead; cover and chill. Let dough come to room temperature before baking.