Preheat oven to 400 degrees F (200 degrees C). Line a baking sheet with parchment paper or aluminum foil.
Place eggplant, cut-side up, on the prepared baking sheet. Sprinkle garlic over each and coat with olive oil; season with salt and pepper.
Bake in the preheated oven until eggplant is mostly tender, 25 to 30 minutes. Add chicken, tomatoes, and Cheddar-Monterey Jack cheese to each eggplant. Continue cooking in the oven until cheese is melted, about 10 minutes. Top each with scallions and basil.