Preheat oven to 350 degrees F. Line a standard muffin tin with paper liners. Place chocolate butter in a heatproof bowl set over (not in) a pan of simmering water. Stir occasionally just until melted, 4 to 5 minutes.
Remove bowl fro heat. Whisk in sugar and salt until mixture is smooth; whisk in eggs to combine. Gently whisk in flour and cocoa just until smooth (do not overmix).
Spoon 1 heaping tablespoon of batter into each lined cup. Place 1 peppermint patty on top, gently pressing into batter. Top with 2 tablespoons batter, covering patty completely. Bake, rotating tin halfway through, until a cake tester inserted halfway in centers (above mint patty) comes out with only a few moist crumbs attached, about 35 minutes. Transfer tin to a wire rack to cool completely before removing cupcakes. Cupcakes can be stored up to 3 days at room temperature in airtight containers.