01 - Heat olive oil in a large pot or Dutch oven over medium-high heat. Add thinly sliced steak and cook for 2–3 minutes until just browned. Remove beef from pot and set aside.
02 - Add butter to the same pot. Once melted, add onion and bell peppers. Sauté for 5–6 minutes until softened. Stir in garlic and cook for 1 minute more.
03 - Sprinkle flour over the vegetables and stir well to coat. Cook for 1 minute to remove the raw flour taste.
04 - Gradually whisk in beef broth, scraping up any browned bits from the bottom. Bring to a gentle simmer.
05 - Add milk, salt, pepper, smoked paprika, and thyme. Simmer for 5 minutes.
06 - Reduce heat to low. Stir in the cooked beef, provolone, and mozzarella. Continue stirring until cheese is melted and soup is creamy.
07 - Taste and adjust seasoning if needed. Serve hot, garnished with extra cheese or chopped parsley if desired.