Preheat the oven to 350 degrees. Line a rimmed baking sheet with parchment paper and set aside.
Put the plantains in a large non-stick pot and cover with cold water. Bring the water to a boil and cook the plantains on medium heat until soft, about 20 minutes.
Transfer the cooked plantains to a large bowl. Reserve the water.
With a potato masher, begin mashing the plantains, and while they are still hot, add the unsalted butter. Season with salt.
Gradually add up to 1 1/2 cups of the cooking liquid to the plantains, mashing, until the mixture achieves a pureed consistency, with only a few small chunks. The mixture will be sticky to the touch, but cohesive enough to handle.
Shape the mixture into patties about the size of the palm of your hands. Make a dent in the middle and add some crumbled queso fresco. Close the patties, ensuring the cheese stays inside.
In a large skillet, heat up about 2 tablespoons of vegetable oil. Fry the patties, in batches using more oil as needed, until both sides are golden brown. The empanadas will have a hard crust on the outside, and should make a noise when tapped with a fork.
Place the fried patties on the prepared baking sheet, and bake them until warm through and the cheese is melted, about 10 minutes.
Serve with hot coffee for breakfast or as an afternoon snack.