Preheat oven to 350˚F (180˚C). Grease 4 (4 inch, 10 centimeter) shallow ramekins or mini springform pans with olive oil.
Bring a large pot of salted water to a boil. Add the spaghetti and cook, stirring occasionally, until barely al dente, just tender.
Drain thoroughly and toss lightly in olive oil. Set aside and allow to cool slightly.
In a medium bowl, whisk together the eggs, Parmesan, parsley, garlic, salt, and pepper.
Toss the mixture quickly with the slightly cooled pasta until thoroughly coated.
Use tongs or a large fork to twirl 1/4 of the pasta and transfer to the prepared ramekin.
Repeat with remaining pasta. Make a large indentation in the center of each pasta twirl and drizzle the tops lightly with olive oil.
Place the ramekins or springform pans on a rimmed baking sheet.
Bake in preheated oven until set and beginning to turn golden on edges, about 15 minutes.
Allow to cool for 10 minutes. If using springform pans, allow to cool for 15-20 minutes, then gently remove from the pans using a thin spatula.
While the pasta nests are baking, warm the tomato sauce in a small saucepan.
Transfer the warmed tomato sauce to the center of each pasta nest.
Place three mozzarella balls in the center of each nest over the tomato sauce and serve immediately.