Heat the oven to 425°F and brush a rimmed sheet pan with olive oil. Mix beef, egg, breadcrumbs, Parmesan, half the garlic, 1/2 teaspoon oregano, 1/2 teaspoon salt and pepper gently in a bowl.
Form 18 equal meatballs, place on the pan and bake for 12 to 15 minutes until lightly browned; they will finish cooking in the slow cooker.
Combine tomatoes, broth, beans, carrots, onion, remaining garlic, oregano and salt in a 6-quart slow cooker. Nestle meatballs into the sauce immediately after baking.
Cover and cook on low for 4 hours, then fold in zucchini gently without breaking the meatballs.
Cover and cook on low for 1 hour more until carrots and zucchini are tender and meatball centers reach 160°F.
Rest uncovered for 10 minutes, then serve 3 meatballs per portion with one-sixth of the beans, vegetables and thick sauce.