In a large bowl, mash together the egg, heavy cream, and the slice of white bread until you form a smooth paste (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.
Mix thoroughly with a spatula or your hands until evenly combined, then shape the mixture into 1-inch meatballs using a heaping tablespoon for each and place them on a baking sheet.
Heat 2 tablespoons olive oil in a large skillet over medium heat. Add the meatballs and brown on all sides, turning occasionally, about 10 minutes; transfer browned meatballs to a paper-lined plate.
In a medium saucepan, heat 1 tablespoon olive oil over medium-high heat until shimmering, then add the minced garlic and cook until fragrant, about 30 seconds.
Add the crushed tomatoes, undrained diced tomatoes, 1 teaspoon dried basil, 1 teaspoon dried oregano, and the optional 1 teaspoon sugar to the saucepan. Bring the sauce to a boil.
Return the browned meatballs to the sauce, reduce heat to medium, and simmer uncovered until the meatballs reach 155°F (about 10–15 minutes) and the flavors meld. Season to taste with salt and pepper.
While the sauce simmers, bring about 4 quarts of water and 1 tablespoon salt to a boil in a pot. Add the spaghetti and cook until al dente, about 10 minutes, then drain well.
Serve the spaghetti topped with meatballs and tomato sauce, and garnish with additional grated Parmesan and minced fresh parsley if desired.