Preheat the oven to 400 degrees F. Line a baking sheet with parchment paper and spray with cooking spray.
Cut off both ends of each plantain, then remove and discard the peels. Place the plantains on the baking sheet and lightly spray with cooking spray. Bake the plantains for 15 minutes, then turn them over and bake until the plantains are lightly golden brown and tender, an additional 15 minutes. Allow to rest at room temperature.
Meanwhile, clip a deep-frying thermometer onto a large, heavy-bottomed skillet. Heat 1 1/2 inches (about 2 1/2 cups) vegetable oil to 375 degrees F.
Fry the plantains in batches until golden, 2 to 3 minutes. Remove with a slotted spoon and drain on paper towels to cool.
Heat the remaining 1/2 cup vegetable oil in a large Dutch oven or stock pot over medium-high heat. Add the onions, bell pepper and garlic and cook, stirring occasionally, until the vegetables have softened, about 5 minutes. Add the tomato paste, then stir and cook another 2 minutes. Add the ground beef, olives, poblano chile and some salt and cook, breaking up the meat, until the meat is entirely browned, 5 to 7 minutes. Add the canned tomatoes with the reserved juices, cumin, paprika, garlic powder and some salt and bring to a boil. Reduce the heat to a simmer and cook for 5 minutes. Fold in half the Cotija, then stir to melt and set aside.
Cut a slit in each of the plantains running the long way. Divide the picadillo chili among the plantains. Sprinkle the tops with remaining Cotija. Garnish with the cilantro and serve.