Make chili: Add onion, celery, and garlic to a food processor and puree until a paste forms. Heat olive oil in a medium heavy-bottomed pot over medium-high heat. Add vegetable puree and cook for 7 minutes, until mixture is fragrant. Add spices and stir until completely incorporated and cook for 30 seconds to toastt he spices.
Reduce heat to medium and add mushrooms. Cook for 10 minutes, or until moisture has totally evaporated and mushrooms brown and begin to caramelize. Add tomato paste and chocolate, stirring until chocolate has melted and mushrooms are completely coated.
Pour in tomatoes, broth, and Worcestershire sauce and bring to a boil. Reduce heat to a simmer and cook uncovered for 1 hour for a looser, Cincinnati-style chili or 1½hours for a thicker chili. Season to taste with salt and pepper.
Make chili dog: Over a medium-high grill or on a grill pan, grill hot dogs until they are heated through and charred. Move to toasted buns and cover with a layer of chili and a sprinkling of raw onion and vegan cheddar (if using).