In a large bowl, mash the egg, heavy cream, and the slice of bread together until a smooth paste (panade) forms.
Add the ground beef, finely chopped onion, 1 tablespoon chopped parsley, 1/4 teaspoon crushed dried oregano, 1 teaspoon salt, and 1/2 teaspoon black pepper to the bowl and mix thoroughly with a spatula or your hands.
Using a heaping tablespoon of mixture at a time, roll into 1-inch meatballs and arrange them on a baking sheet.
Heat 2 tablespoons olive oil in a large skillet over medium heat. Add the meatballs and cook, turning occasionally, until browned on all sides, about 8–10 minutes. Transfer browned meatballs to a paper-lined plate.
In a medium saucepan over medium-high heat, warm 1 tablespoon olive oil until shimmering. Add the minced garlic and cook, stirring, about 30 seconds until fragrant.
Add the crushed tomatoes, undrained diced tomatoes, 1 teaspoon dried basil, 1 teaspoon dried oregano, and the optional 1 teaspoon sugar; bring the sauce to a boil while stirring. Reduce heat to medium.
Return the browned meatballs to the sauce, simmer uncovered for 10–15 minutes until flavors meld and meatballs reach 155°F (about 10–15 minutes). Taste and season the sauce with salt and pepper as needed.
While the sauce simmers, bring 4 quarts of water and 1 tablespoon salt to a boil. Cook 1 pound spaghetti until al dente, about 10 minutes, then drain well.
Serve the spaghetti topped with sauce and meatballs and garnish with additional grated Parmesan and minced fresh parsley if desired.