In a large bowl, mash the egg, heavy cream, and white bread slice together until a smooth paste forms (an apanade).
Add the ground beef, finely chopped onion, grated Parmesan, chopped parsley, 1/4 teaspoon dried oregano, 1 teaspoon salt, and 1/2 teaspoon black pepper to the bowl and mix thoroughly with a spatula or your hands until combined.
Using about 1 heaping tablespoon of mixture per meatball, roll into 1-inch balls and arrange them on a baking sheet.
Heat 2 tablespoons olive oil in a large skillet over medium heat. Add the meatballs and brown, turning occasionally, about 8–10 minutes; remove browned meatballs to a paper-lined plate.
In a medium saucepan, heat 1 tablespoon olive oil over medium-high until shimmering. Add the minced garlic and cook until fragrant, about 30 seconds.
Carefully add the crushed tomatoes and diced tomatoes (undrained) to the saucepan, then stir in 1 teaspoon dried basil, 1 teaspoon dried oregano, and 1 teaspoon sugar (optional). Bring the sauce to a boil over medium-high heat.
Add the browned meatballs to the sauce, reduce heat to medium, and simmer uncovered until the flavors meld and meatballs reach 155°F internally, about 10–15 minutes; season with additional salt and pepper to taste.
While the sauce simmers, bring 4 quarts of water and 1 tablespoon salt to a boil in a large pot. Add the spaghetti and cook until al dente, about 10 minutes; drain well.
Serve the drained spaghetti topped with meatballs and tomato sauce and garnish with additional grated Parmesan and minced fresh parsley if desired.