Heat the vegetable oil in a large skillet over medium-high heat.
Sauté the ground beef with the onions and peppers for 5-7 minutes, seasoning with salt, pepper, cumin, and chili.
Add the garlic and sauté for another minute.
Add the tomato paste, crushed tomatoes, and broth, and cook for about 10 minutes until the chili thickens. Adjust the seasoning to taste.
Slice the grilled sausages or hot dogs lengthwise, making sure that they are still attached at the bottom. Top with spoonfuls of the beef chili and serve hot, optionally topped with sour cream, avocado, cilantro, and lime wedges on the side.