Place an oven rack in the center of the oven. Preheat the oven to 350 degrees F. Butter the bottom and sides of four 4-inch-diameter springform pans.
For the pasta: Bring a large pot of salted water to a boil over high heat. Add the pasta and cook until tender but still firm to the bite, stirring occasionally, 8 to 10 minutes. Drain the pasta and place in a large bowl. Add the spinach, cheese, oil and 1/2 teaspoon salt. Toss well until the spinach is wilted. Set aside to cool.
For assembly: Beat the Parmesan, milk, salt and egg together in a large bowl until smooth. Pour the sauce over the spaghetti mixture until well combined. Twirl 1/2 cupfuls of the spaghetti mixture around tongs or a large fork and transfer to the prepared pans. Make an indentation in the middle of each nest of pasta and drizzle with olive oil. Bake 15 minutes. Let cool 10 minutes before removing the sides from the pans. Remove the bottoms of the pans from the nests using a thin metal spatula.
Spoon 1/2 cup of the marinara sauce on each of 4 serving plates. Place the spaghetti nests on the sauce. Place 3 balls of the mozzarella in each nest and serve.