How to Make It
Preheat the oven to 400°. Line a rimmed baking sheet with foil. Place the squash cut side up on the prepared baking sheet. Season generously with salt and pepper and then drizzle with 1 tablespoon of the olive oil. Roast, cut side down, for about 1 hour, until the squash can be pierced easily with a knife. Let the squash stand at room temperature until cool enough to handle, about 15 minutes.
Using a fork, scrape the spaghetti squash into a colander set over a large bowl to drain for 15 minutes. Transfer the drained squash to a large bowl. Add the butter and 1/2 cup of the Parmigiano to the squash and toss gently. Season generously with salt and pepper.
Put the flour, eggs and panko in 3 separate shallow bowls. Season the chicken breast halves with salt and pepper and coat them with flour, tapping off the excess. Dip in the beaten eggs and then in the panko, pressing to help the crumbs adhere. Transfer to a large plate.
In a large skillet, heat 1/4 cup of the vegetable oil until shimmering. Add 2 of the breaded chicken breast halves to the skillet and cook over moderately high heat, turning them once, until browned on both sides, about 3 minutes. Transfer to a wire rack and season with salt. Carefully discard the oil in the skillet and wipe clean. Repeat with the remaining chicken and vegetable oil.
Lightly grease a 9-by-13-inch baking dish with olive oil. Spread the cooked spaghetti squash in the bottom of the prepared baking dish. Spoon 1 1/2 cups of the marinara sauce over the squash. Arrange the browned chicken breast halves on top of the marinara sauce in a single layer. Spoon the remaining 1cup of the marinara sauce over the chicken. Sprinkle the chicken with the mozzarella and 1/4 cup of the Parmigiano.
Bake for 30 to 35 minutes, until the sauce is bubbling and the cheese is beginning to brown. Remove from the oven and let stand at room temperature for 10 minutes. Sprinkle with parsley and the remaining the remaining 1/4 cup of Parmigiano and serve.