Using a Dutch oven, combine the ground chuck, sausage, onions, and garlic. Cook over medium heat until the meat is browned, stirring the meat to break it up as it cooks. Drain the pan drippings. Stir in the chili powder, flour, sugar, oregano, salt, and stir well. Cover pot and simmer for 30 minutes, stirring occasionally. Add the tomatoes and beans, and simmer for another 20 to 30 minutes.
Preheat a grill. Butterfly, or gently slice hot dogs lengthwise, making sure not to cut all the way through. Grill butterflied hot dogs until done, approximately 5 minutes. Place the buns on the grill to warm them. Place hot dogs in the bun. Put the hot dog on the plate with split side facing up. Ladle chili over and top with cheese.