For the sauce: Heat the oil in a medium saucepan over medium heat. When it shimmers, add the onion and garlic and season with salt and freshly ground black pepper. Cook, stirring occasionally, until the onions are soft and golden, about 5 minutes. Add the tomatoes and sugar, bring to a simmer, and cook, stirring frequently, until slightly thickened, about 10 minutes. Taste, adjust seasoning, if needed, and set aside.
For the dumplings: Bring a medium pot of heavily salted water to boil over high heat. Add the arugula and blanch, about 30 seconds. Remove arugula with a slotted spoon, drain, and squeeze out as much moisture as possible. Chop finely and set aside. Return the pot to the stove, and bring to a simmer over low heat.
Combine the milk and the eggs in a large bowl and beat until well combined. Add the bread and mix until soaked through. Stir in the chopped arugula and mix thoroughly. Add the remaining ingredients, except the mozzarella, and stir until just evenly combined.
Preheat the broiler.
Shape the dough into tablespoon-sized dumplings (you should have about 30). Drop about 10 dumplings into the simmering water and cook until they float to the surface and are cooked through, about 5 minutes. Remove with a slotted spoon to a colander and repeat with remaining dumpling dough.
Once all the dumplings are cooked, transfer to an oven-safe serving dish, spoon the tomato sauce over the top and cover with the mozzarella. Broil until the cheese is bubbly and lightly browned. Serve immediately.