Press Sauté button and add butter and oil to the Instant Pot. Once melted, add the onion and carrots and sauté until onions are translucent and carrots have softened, about 3 - 4 minutes.
Add garlic and cook for 1 minute, then stir in flour until well combined. Add broth and stir until smooth.
Add the broccoli and leave on top of the soup. Place the lid on the Instant Pot. Press the Cancel button to turn it off, then press the Manual button and set for 8 minutes.
After 8 minutes, use the Quick Release and allow the steam to escape. Open the lid and turn off the Instant Pot.
Stir in the paprika and Dijon mustard. Add the cheese and stir until combined. Stir in the milk or half and half. Taste and season as needed with salt and pepper. Garnish with fresh parsley or fresh thyme leaves and serve.