Preheat your oven to 350°F. Grease an 8×8-inch metal baking pan with butter, then line it with parchment paper, leaving overhang on two opposite sides for easy removal. Lightly butter the parchment as well to ensure nothing sticks.
In a medium saucepan over low heat, melt the butter and milk chocolate together, stirring occasionally. Once most of the chocolate has melted, remove from heat, stir in the espresso powder, and set aside to cool slightly.
In a large mixing bowl, beat the cane sugar, brown sugar, eggs, vanilla paste, and salt with a hand mixer on medium speed for about 2 minutes, until the mixture is slightly thick and glossy. Reduce the speed to low and slowly pour in the melted chocolate mixture along the side of the bowl to prevent splattering. Mix until fully incorporated.
In a separate bowl, whisk together the flour and cocoa powder. Add to the wet mixture and mix on low speed just until no dry spots remain.
Chop the peanut butter cups into bite-sized pieces. Gently fold half of them into the batter, then pour the batter into the prepared pan and smooth it into an even layer.
Bake for 15 minutes, or until the top just starts to set. Carefully remove the pan from the oven and tap it firmly on the counter once or twice (placing a towel underneath if needed). This helps create cracks on the surface and keeps the brownies fudgy.
Scatter the remaining chopped peanut butter cups over the surface, then return the pan to the oven and bake for another 12–14 minutes, or until the edges are set and the center no longer jiggles when gently shaken.
Let the brownies cool completely in the pan at room temperature, about 2 ½ hours. Once fully set, lift them out using the parchment overhang and slice into 9 squares. Store leftovers in an airtight container at room temperature for up to 4 days.