In a medium saucepan, brown the butter: cook over medium heat until the butter melts then turns caramel brown in color, about 5-7 minutes. The butter will foam and bubble. Stir often. Transfer into a large bowl (I recommend plastic, glass, ceramic, or bamboo. DO NOT use a stainless steel for faster cooling). The brown butter will contain sediment that sinks to the bottom.
Cool browned butter for 10-15 minutes, at room temperature. Then add brown sugar and sugar. With an electric hand mixer, beat on low speed until the sugars and butter have combined, about 1-2 minutes. It should look like muddy sand.
For thin + crispy cookies: add the vanilla and ROOM TEMPERATURE eggs and beat until just combined.
For thick + chewy cookies: add the vanilla and eggs (the eggs can be cold), one at a time, and beat until just combined. Then increase the hand mixer speed to medium and beat until lighter in color and thicker. This doesn't take long -- an additional 30-60 seconds at most. Don't overmix!
Add flour, salt, and baking soda. Mix on low until just combined.
Add chocolate chips and toffee bits. Switch to a hand held spatula. Mix until fully combined. The cookie dough will look very moist.
Set aside for 30 minutes, at room temperature. The batter will thicken slightly.
Meanwhile, preheat the oven to 375F/190C and line sheet pans with parchment paper.Using a medium cookie scoop, scoop cookie dough onto the sheet pan. For smaller cookies, use a medium cookie scoop which measures 2 Tbsp. For larger cookies, use a large cookie scoop which measures 3 Tbsp. Do not flatten as the cookies will spread on their own. Sprinkle with flaky sea salt.
Bake until edges are golden brown and firm but the center is still soft, about 7-11 minutes. The cookies will be puffy when they first come out of the oven but will "collapse" as they cool. Let the cookies sit on the baking sheet for 10 minutes before transferring to a wire rack. Cool and enjoy.