Boil plantains in a large pot of water until tender, 20 to 30 minutes; drain.
Peel plantains while still hot and mash in a bowl until smooth.
Pour mashed plantains into a saucepan; add 1/2 cup sugar, 2 cinnamon sticks, 1 teaspoon vanilla extract, and 1/4 teaspoon salt. Cook and stir over medium heat until sugar is dissolved, about 5 minutes. Refrigerate until cooled, about 20 minutes.
Whisk 1/2 cup milk and cornstarch together in a small bowl until smooth.
Combine 2 cups milk, 2 tablespoon sugar, lemon zest, 1 cinnamon stick, 1 teaspoon vanilla extract, and 1/4 teaspoon salt in a saucepan over medium heat; bring to a boil. Add cornstarch mixture and cook, stirring constantly, until filling thickens, 3 to 5 minutes.
Pinch off a piece of the plantain mixture and roll into a ball the size of a golf ball. Flatten into a disc and place 1 tablespoon of filling in the center; fold in half and press edges to seal into an empanada. Repeat with remaining plantain mixture and filling.
Heat oil in a deep skillet. Fry empanadas in batches until golden brown, about 5 minutes per side. Drain on paper towels.