Heat the oven to 400°F. Meanwhile, place 1 stick unsalted butter, 2 minced garlic cloves, 4 teaspoons whole-grain Dijon mustard, 4 teaspoons prepared horseradish, 1 tablespoon lemon juice, ½ teaspoon smoked paprika, ½ teaspoon kosher salt, ¼ teaspoon black pepper, and ¼ teaspoon red pepper flakes if using in an 8x8-inch baking pan. Place in the oven while it is heating until the butter melts, about 7 minutes.

Remove the baking pan from the oven. Stir the mixture to make sure everything is evenly combined and so that it coats the sides of the baking pan. Transfer 1 tablespoon of the mixture to a small bowl, preferably microwave-safe.

Whisk 2 ½ cups all-purpose flour, 1 tablespoon baking powder, 1 ½ teaspoons kosher salt, and 1 teaspoon baking soda together in a large bowl. Add 4 ounces shredded sharp cheddar cheese and 1 tablespoon finely chopped fresh chives, and toss to combine. Stir in 1 ½ cups cold buttermilk with a flexible spatula until a shaggy, sticky dough forms and there are no pockets of dry flour left.

Transfer the dough into the baking pan. Using the spatula, press the dough evenly into the pan until it touches all the sides (the dough will be swimming in butter mixture). Using a metal bench scrape or knife, cut the dough all the way down to the bottom of the pan in a 3x3 grid to make 9 squares. (The squares will still be touching each other.)

Bake until the biscuits are golden-brown and have puffed up significantly, and a tester inserted into the center biscuit comes out clean, 25 to 27 minutes. Immediately cut along the scores again to separate the biscuits, then brush the top with the reserved cowboy butter (microwave the butter if it solidified to melt again if needed). Let cool for at least 15 minutes in the pan before serving.