In a bowl , season ground beef with salt, pepper and cumin. Reserve.
In a large saucepan over medium-high heat, add the sausage and just enough water to cover. Once the water comes to a boil, remove pan from heat. You will notice that the sausages have nearly doubled in size (plumpness technique). Reserve.
In a medium stainless-steel saucepan over medium-high heat, heat the oil. Sauté both the onion and bell pepper. When the onion becomes translucent, add the garlic and let it cook, stirring constantly, for 30 seconds.
Add the ground beef and let brown. Stir in the tomato paste. Then add either water or broth/stock and thyme, and bring to a boil.
When it begins to boil, reduce heat to medium-low and let the meat cook, partially covered, for about 10 minutes, stirring occasionally.
Remove from heat, stir in the lime juice, and cover the pan.
To assemble the hot dog, put a warm or hot sausage link into each roll/bun, and then on top of it spoon about 2-4 tablespoons of the prepared, hot ground beef. Top with shredded mozzarella cheese (only assemble the hot dogs right before serving, in order to prevent the bread from becoming soggy).
Mustard, ketchup, and mayonnaise are the preferred choices for condiments in Brazil. The hot dogs can be served with the following accompaniments: steamed corn, steamed peas, pico de gallo, grated Parmesan cheese, shoestring potatoes, shredded lettuce, chopped parsley, chopped olives, grated carrots, mashed potatoes, requeijão cheese, etc.. Typical accompaniments vary from one region of Brazil to the next.