Dice the onion into small pieces and mince the garlic. Cut the broccoli into small 1/2-inch florets—uniform sizing ensures even cooking under pressure. Shred or julienne the carrot into 1-inch matchsticks so it cooks at the same rate as the broccoli. Measure out the broth, and combine the salt, black pepper, red chili flakes, and ground nutmeg in a small bowl so they're ready to add.

Set the Instant Pot to SAUTE mode and add 3 tablespoons of butter. Once melted and foaming, add the diced onion and cook for 1-2 minutes, stirring occasionally, until softened and fragrant. Add the minced garlic and cook for an additional 30 seconds until aromatic—this short cooking time builds flavor without burning the garlic.

Stir in the prepared broccoli florets, carrot matchsticks, and seasoning mixture (salt, pepper, chili flakes, nutmeg) from Step 1. Pour in the broth and stir well to combine. Cancel SAUTE mode, seal the lid, and cook on high pressure for 5 minutes. While the pot comes to pressure and cooks, this allows the vegetables to become tender and infuse the broth with flavor.

Once the 5-minute cook time is complete, carefully turn the pressure valve to VENTING and allow the steam to release naturally—do not use quick release, as this can break up the delicate vegetables. While the pressure releases (this takes about 5-10 minutes), prepare the cheese roux in a separate small saucepan: melt 2 tablespoons of butter over medium heat, then whisk in 2 tablespoons of flour, stirring constantly for about 1 minute to cook out the raw flour taste. Gradually whisk in 1 cup of milk, stirring until the mixture becomes smooth and thickens (usually 2-3 minutes). I like to do this simultaneously with pressure release to save time—it's a great multitasking moment.

Once the pressure has fully released, open the Instant Pot and set it back to SAUTE mode. Stir in the thickened milk mixture from Step 4, then add the sharp cheddar cheese and stir constantly for 1-2 minutes until the cheese is completely melted and the soup becomes creamy and smooth. The sharp cheddar gives better texture and richness than milder varieties. Simmer for an additional 2 minutes to ensure everything is well combined and heated through.

Taste the soup and adjust seasoning with additional salt if needed. Cancel SAUTE mode and ladle the soup into bowls. Serve hot, garnishing with extra sharp cheddar, freshly cracked black pepper, or a pinch of red chili flakes for added flavor and visual appeal.