Set your Instant Pot to 'Sauté' mode. Add the butter and let it melt, then stir in the diced onion, cooking for 2-3 minutes until softened.
Add the minced garlic and shredded carrot, sautéing for another minute until fragrant.
Add the chopped broccoli florets, vegetable broth, kosher salt, pepper, smoked paprika (if using), and nutmeg. Stir to combine.
Secure the Instant Pot lid and set valve to 'Sealing.' Cook on high pressure for 5 minutes.
Once finished, do a quick release carefully, using a kitchen towel to avoid steam burns.
For a chunkier soup, use a potato masher to break up the broccoli. For smooth soup, use an immersion blender directly in the pot until desired consistency is reached.
In a small bowl, whisk together the milk and flour until smooth. Stir this mixture into the soup.
Set Instant Pot to 'Sauté' again, bringing the soup to a gentle simmer for 3-4 minutes to thicken, stirring frequently.
Turn off the Instant Pot. Gradually stir in the shredded cheddar cheese until melted and fully incorporated.
Taste and adjust seasoning if needed. Serve hot, garnished with extra cheddar if desired!