Step 1

Preheat oven to 350°. Toast peanuts on a rimmed baking sheet until browned and fragrant, 6–8 minutes. Let cool, then coarsely chop.

Step 2

Meanwhile, cook butter in a medium skillet over medium heat, swirling pan often, until butter foams, then browns, 4–6 minutes. Transfer brown butter with solids to a medium bowl; chill, stirring every 5 minutes, until cooled and beginning to solidify, about 15 minutes.

Step 3

Whisk flour, kosher salt, baking powder, and baking soda in another medium bowl. Using an electric mixer on high speed, beat egg, egg yolk, granulated sugar, brown sugar, and ¼ cup water in a large bowl until mixture is light and falls back on itself in a slowly dissolving ribbon, about 2 minutes.

Step 4

Reduce mixer speed to low and gradually add dry ingredients, then increase speed to high and beat 1 minute to develop gluten and hydrate flour. Add peanut butter, toasted peanuts, and brown butter and mix on low speed, scraping down sides of bowl as needed, until fully incorporated, about 30 seconds. Let dough rest at room temperature until slightly firmed up, about 10 minutes.

Step 5

Portion dough into 20 balls (about 2 heaping Tbsp. each) and transfer to 2 parchment-lined rimmed baking sheets. Using a fork dipped in cold water, flatten cookies to ½" thick, making a crosshatch pattern on the top of each. Chill, covered, 1 hour.

Step 6

Arrange racks in upper and lower thirds of oven; place a dry, medium skillet on the bottom rack, then increase oven temperature to 375°.

Step 7

Sprinkle cookies with demerara sugar and sea salt. Working quickly, transfer 1 sheet of cookies to upper rack, then carefully pour 1 cup water into hot skillet; water will bubble and sizzle violently. Bake cookies until browned and edges are crisp, 12–15 minutes. Let cool slightly on baking sheet, then transfer to a wire rack and cool completely. Repeat with second sheet.

Step 8

Do Ahead: Cookie dough can be shaped and frozen 1 month ahead. Freeze in an airtight container. Let rest at room temperature 1 hour before baking.