In a medium skillet, heat the olive oil over medium-high heat. Add the onions and garlic and cook, stirring often, for 30 seconds.
Add the broth, tomatoes, tomato sauce, vodka, parsley, salt and sugar to the slow cooker; add the sauteed onions and garlic. Cover and cook on high heat to stew the tomatoes, 5 hours. Discard the parsley sprigs.
Using an immersion blender or a food processor, puree the soup in batches (be careful-the liquid is hot). Stir in the half-and-half. Cook in the covered slow cooker on low heat until heated through, 20 to 30 minutes.