Put a rack in the center of the oven and heat to 375°F. Line a large rimmed baking pan with parchment or a nonstick liner and set aside.

Put the breadcrumbs and milk in a large mixing bowl, stir to blend, and set aside for 5 minutes. With your hands, squeeze and mash the breadcrumbs so that they make a smooth paste. Add the egg and whisk, then add the Parmigiano, ketchup, parsley, oregano, thyme, half the onion, half the garlic, 2 teaspoons salt, and 1/2 teaspoon pepper. Stir to blend.

With your hands, break the beef into small chunks and add to the bowl. Mix gently but thoroughly; overmixing will make the meatballs tough and dry. When all the ingredients are evenly combined, shape the mixture into 6 balls and arrange on the baking sheet.

Bake until an instant-read thermometer reads 165°F at the center of a meatball, 40 to 45 minutes.

While the meatballs are baking, make the sauce. Heat the olive oil in a medium saucepan over medium heat. Add the remaining half onion and 1 teaspoon salt and increase the heat to medium-high. Cook, stirring frequently, until translucent, about 3 minutes. Add the remaining

half garlic and cook until aromatic, about 1 minute. Add the tomatoes, bring to a boil, and lower the heat to maintain a vigorous simmer.

Cook, stirring occasionally, until reduced to a thick sauce, 10 to 15 minutes. Taste the sauce and add a pinch or two of sugar if it seems too tangy, and season with more salt if necessary. Stir in the basil and keep warm until the meatballs come out of the oven.

Spoon the sauce generously over the meatballs and bake for another 5 to 10 minutes to blend the flavors. Garnish with a sprinkling of Parmigiano.