1/2 cup (4 ounces) unsalted butter, at room temperature
3/4 cup cane sugar
1/2 cup firmly packed light muscovado sugar or light brown sugar
1 cup creamy peanut butter
1 extra-large egg, at room temperature
1 tablespoon whole milk
1 teaspoon pure vanilla extract
3/4 cup unbleached all-purpose flour
1/2 teaspoon kosher salt
1/2 teaspoon sea salt flakes
1/2 teaspoon baking powder
1/4 teaspoon baking soda
1 cup cane sugar
Shards of Peanut Brittle, store-bought or homemade, for garnish (optional)
In the bowl of a stand mixer fitted with the paddle attachment, mix the butter briefly on medium speed for 5 to 10 seconds. Add the sugars and beat until the butter mixture is aerated and pale in color, approximately 4 minutes. Scrape the sides and bottom of the bowl with a rubber spatula to bring the batter together. Add the peanut butter and mix on medium speed to combine thoroughly, approximately 1 minute.
Crack the egg into a small cup or bowl and add the milk and vanilla.
In a bowl, whisk together the flour, salts, baking powder, and baking soda.
On medium speed, gradually add the egg, milk, and vanilla to the butter mixture. Mix for 5 seconds. Scrape the sides and bottom of the bowl with a rubber spatula to bring the batter together. Mix on medium speed for 20 to 30 seconds to make nearly homogeneous.
Add the dry ingredients all at once and mix on low speed until the dough comes together but still looks shaggy, approximately 30 seconds. Scrape the sides and bottom of the bowl to bring the batter together. Mix for another 10 seconds on medium speed. Remove the bowl from the stand mixer. With a plastic bench scraper, bring the dough completely together by hand.
Transfer the dough to a sheet of plastic wrap and wrap tightly. Refrigerate overnight.
Heat the oven to 350°F and line a couple of half sheet (13x18-inch) pans with parchment paper.
To make the coating: Put the sugar for coating the cookies in a bowl, ensuring there is plenty of room to roll the dough in the sugar. Using a 3/4-ounce (1 1/2-tablespoon) ice cream scoop, portion the dough into 12 mounds and roll into balls. Gently coat each ball completely with the sugar. Evenly space the balls on a prepared sheet pan. Press a couple of peanut brittle shards into the top of the rounds.
Bake the cookies for 8 minutes. Rotate the pan and bake until the tops are barely set and the edges are lightly golden, 4 to 5 minutes. They will firm up as they cool, but they will still be chewy. For crispy cookies, bake for