Drain grease from pan and melt butter over medium heat. Add shallot and cook 2-3 minutes until tender. Add garlic, red pepper flakes, and oregano to pot and saute 1 additional minute until fragrant. Deglaze pan with vodka, scraping up any browned bits.
Add tomato paste, whole tomatoes, basil leaves, and sugar to pot. Bring sauce to a simmer over medium-high heat, then reduce heat to medium-low, cover, and let simmer 20-30 minutes until slightly reduced, stirring and smashing tomatoes occasionally.
Discard basil leaves, transfer sauce to a blender, food processor or use an immersion blender and blend on high speed until sauce is smooth or to your desired level of chunkiness.
Pour sauce back into pot and add meatballs, orzo, and chicken broth. Stir until evenly combined. Bring to a simmer over medium heat, then reduce heat to medium-low and allow to simmer 10-15 minutes, stirring occasionally until most of the liquid is reduced, orzo is tender, and meatballs registers 165F.
Season orzo with salt and pepper to taste and add an additional splash of chicken broth if needed to loosen sauce. Serve meatballs and orzo warm with grated Parmesan on top and French bread for dipping. Enjoy!