1In medium fry pan over medium-high heat, heat 1 tablespoon of the vegetable oil. Add chopped plantain and sauté until golden and slightly softened, about 5 minutes. Transfer plantains to a bowl and set aside.
2Heat remaining 1 tablespoon oil in the sauté pan and add onion. Sauté until golden, about 4 minutes. Add beans, cilantro, cumin, and cayenne and sauté until mixture is heated and flavors are blended, about 5 minutes. Add lime juice and season with salt. Remove from heat.
3Using a potato masher, mash bean mixture until it forms a coarse paste. Transfer to a bowl and set aside to cool. Add plantains to bean mixture and stir to combine.
4Line two baking sheets with parchment paper. In a small bowl, whisk together egg and milk to make egg wash.
5Spread one sheet of puff pastry on a lightly floured surface. Dust pastry lightly with flour and, using a rolling pin, roll into a 14-inch square.
6Using a sharp paring knife, trim dough edges. Cut dough into four equal squares. Using a pastry brush, brush edges of each square lightly with egg wash. Place 1/4 cup of cooled filling on each square and sprinkle with 2 tablespoons of queso fresco and 1 1/2 teaspoons of butter. Fold dough in half to form a triangle. Seal edges by pressing them together with the tines of a fork. Transfer to a baking sheet. Poke a few holes in the top of each empanada with the tip of a paring knife. Repeat with remaining sheet of puff pastry and remaining filling.
7Cover the baking sheets with plastic wrap and refrigerate for 15 to 30 minutes to firm slightly. Meanwhile, heat the oven to 400°F.
8Remove empanadas from the refrigerator and remove plastic wrap. Brush top of each empanada with egg wash.
9Bake until golden brown and flaky, 20 minutes. Transfer to a wire cooling rack to cool slightly. Serve warm, with tomatillo salsa or sour cream on the side.