Coat portobello mushrooms with about 1/4 cup olive oil on both sides; arrange mushrooms gill-side up on a baking sheet.
Heat 3 tablespoons olive oil in a large skillet over high heat. Cook and stir onion and garlic in hot oil until onion begins to turn translucent, 3 to 5 minutes.
Stir turkey into onion mixture and reduce heat to medium-high; season with salt. Continue cooking and stirring until turkey is no longer pink, 5 to 7 minutes.
Drain turkey, reserving about 1 tablespoon grease in the skillet.
Stir tomatoes and pesto sauce into the turkey; simmer, stirring occasionally, until the sauce is heated through, 5 to 7 minutes.
Place a slice of mozzarella cheese in the center of each mushroom cap and sprinkle evenly with Parmesan cheese.
Divide turkey pesto sauce evenly over the mushroom caps.
Bake in the preheated oven until cheese has melted, 10 to 15 minutes.
