Warm the olive oil in a big pot on medium heat. Put the minced meat in the pan and cook it until it turns brown, using a spoon to break it into smaller pieces. Remove any extra grease if needed.
Add the chopped onion, garlic, carrot, and celery. Cook the vegetables for 4-5 minutes until they are soft.
Add the beef broth and heat it until it boils. Put the elbow pasta in and lower the heat to a low simmer. Cook for around 8-10 minutes, stirring occasionally, until the pasta is firm to the bite.
In a little bowl, mix the milk, cream, and flour until it's smooth. Pour the mixture slowly into the pot while stirring constantly to prevent clumps from forming. Cook for 2-3 minutes until the soup becomes a bit thicker.
Add the grated cheddar and mozzarella cheeses, smoked paprika, mustard powder, salt, and pepper. Keep stirring until the cheese is completely melted and the soup is smooth.
Put the soup in bowls and add fresh parsley on top if you like. Enjoy it hot with crunchy bread or crackers for dipping.