Preheat the oven to 350 degrees F. Lightly oil a 9-by-13-inch baking dish. Scrape out the gills of the portobellos with a spoon, then halve the mushrooms horizontally to make 6 thin rounds.
Heat the olive oil in a saucepan over medium-high heat. Add the red pepper flakes and garlic; cook 1 minute. Reduce the heat to low, add the tomatoes and basil and cook until the garlic is soft, about 15 more minutes. Transfer to a food processor and puree until smooth. Season with salt and pepper.
Combine the panko, 1/2 cup parmesan, 1 tablespoon parsley, 1 teaspoon salt and 1/2 teaspoon pepper in a shallow bowl. Whisk the eggs and 2 tablespoons cold water in another bowl. Put the flour in a third bowl.
Dredge the mushrooms in flour, shaking off the excess. Dip in the eggs and then in the panko mixture, pressing to coat both sides. Heat 1/2 inch peanut oil in a large skillet over medium-high heat. Working in batches, fry the mushrooms until golden, about 2 minutes per side. Drain on paper towels.
Spread a layer of the tomato sauce in the prepared baking dish. Add the fried mushrooms, then cover with the remaining tomato sauce. Top with the mozzarella and the remaining 1/2 cup parmesan. Bake until browned, 20 to 25 minutes. Sprinkle with the remaining 1 tablespoon parsley.