Cook butter in a medium saucepan over medium heat, stirring often, until it foams, and then browns, 5–8 minutes. Let cool slightly.
Meanwhile, whisk flour, baking soda, and kosher salt in a medium bowl.
Add brown sugar, granulated sugar and browned butter to an electric mixing bowl. On medium speed, beat until incorporated, about 1 minute. Add eggs and vanilla and beat until mixture lightens and begins to thicken, about 30 seconds.
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 a rubber spatula. Let dough sit at room temperature or in the fridge for at least 30 minutes to set up. Dough will look very wet/loose at first, but will thicken as it sits.
Preheat oven to 375°. Using an ice cream scoop, portion out equal balls of dough and place (I did 5 at a time) on a parchment-lined baking sheet, spacing a few inches apart. 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 because they are v soft and then transfer to a wire rack and let cool completely. Repeat with remaining dough or save the dough and bake later!