Melt butter in a large pot over medium heat. Add onion and garlic; cook and stir until softened, about 5 minutes. Add tomatoes, celery, vermouth, lemon juices, lemon slices, hot sauce, thyme, paprika, cayenne, and black pepper. Simmer until flavors combine, about 20 minutes.
Stir mussels into the pot. Simmer, covered, until mussels have opened, about 2 minutes.
