MEATBALLS: Combine beef, garlic, parsley, breadcrumbs, mayonnaise, milk, smoked paprika, and salt in a bowl. Using your hands, gently mix all ingredients to incorporate. Shape mixture into 2 tablespoon–size balls. Poke a hole in each ball and insert 1 cheese cube. Re-form meatballs to seal in cheese. Refrigerate meatballs for 30 minutes.
Warm olive oil in a large, wide skillet over medium-high heat. Sear meatballs, in batches if necessary, until browned on all sides, about 10 minutes total (they will not be cooked through). Transfer meatballs to a platter and set aside; do not wipe out skillet.
SAUCE: Add olive oil to same pan used for the meatballs. Once oil is warm, add onion and sauté until translucent, about 5 minutes. Add garlic and cook for 1 minute. Add paprika, anchovy paste, and sugar, stirring to combine. Cook for 1 minute, then add wine, stirring quickly to scrape up browned bits from bottom of pan. Add tomatoes and bring mixture to a boil.
Add meatballs to sauce and reduce heat to medium-low. Cover and simmer until meatballs are cooked through, about 30 minutes. Serve immediately.