Preheat the oven to 350 degrees F. Grease two 12-cavity mini doughnut pans with olive oil.
Bring a large pot of salted water to a boil. Add the spaghetti and cook according to the package instructions until al dente. Drain the pasta and transfer it to a large bowl. Set aside.
While the pasta is cooking, melt the butter in a medium saucepan over medium heat. Add the 2 tablespoons olive oil and the crushed red pepper and heat until the oil is hot. Add the onion and cook, stirring occasionally, until golden, 4 to 6 minutes.
Add the vodka to the onions and let simmer until it reduces slightly and the alcohol cooks off, 1 to 2 minutes. Add the tomato sauce and heavy cream. Stir and let simmer over medium-low heat until the sauce thickens slightly, 8 to 10 minutes. Remove from the heat and then stir in the 1 cup Parmigiano.
Pour the vodka sauce over the pasta and toss to coat, then toss with the mozzarella and egg.
Spoon an equal amount of the pasta into each cavity of the prepared doughnut pans. Bake until golden brown and the doughnuts feel firm, 30 to 35 minutes.
Remove the doughnuts from the pans, sprinkle with the parsley and remaining 2 tablespoons Parmigiano and serve.