Beat the softened butter on medium speed for 2 minutes until it becomes light and fluffy—this incorporates air into the dough, which gives the cookies a better texture. Add the sweetened condensed milk and beat for another minute until fully combined and smooth. The condensed milk acts as both sweetener and binder, creating a denser, chewier cookie than traditional butter-sugar recipes.
In a separate small bowl, sift together the flour, cocoa powder, vanilla extract, and salt. Sifting the cocoa powder is important—it breaks up any clumps and ensures even chocolate distribution throughout the dough. Add this dry mixture to the creamed butter mixture and fold together on low speed until just combined. I like to finish this by hand with a wooden spoon, which prevents overmixing that can make cookies tough.
Refrigerate the dough for 30 minutes (or 15 minutes in the freezer if you're short on time). The chill firms up the dough so it won't stick when rolling and helps prevent the cookies from spreading too much during baking. Once chilled, place the dough between two sheets of parchment paper and roll it out to about 1 centimeter (roughly 3/8 inch) thickness. Rolling between paper prevents the cocoa from staining your surface and eliminates the need for extra flour.
Peel away the top layer of parchment paper and use cookie cutters to cut the dough into your desired shapes. Carefully transfer each cookie to a parchment-lined baking sheet, spacing them about 1 inch apart. I recommend leaving some space between cookies even though these don't spread much—it ensures even baking and makes them easier to remove.
Preheat your oven to 320°F and bake the cookies for 5-8 minutes, watching carefully as they approach the end of baking—the cookies won't look dramatically different when done, but they'll feel slightly firm to the touch. Remove from the oven and let them rest on the baking sheet for 2-3 minutes to set, then transfer to a cooling rack. This cooling method prevents them from breaking apart while still allowing air circulation underneath so the bottoms don't get soggy.