Turn your Instant Pot to the sauté setting and melt the first tablespoon of butter. Add the diced onion, celery, and carrots, stirring occasionally until they start to soften. This only takes a few minutes and really builds the soup's flavor foundation.
Pour in the chicken broth and add your chopped broccoli florets right on top of the softened vegetables. Give everything a quick stir so nothing sits at the bottom of the pot before you seal the lid.
Secure the lid, set the valve to sealing, and cook on high pressure for about 2 minutes. Once the timer ends, do a quick release so the broccoli doesn't turn mushy from sitting in residual heat.
While the pot vents, melt the remaining butter in a small saucepan, whisk in the flour, and cook briefly until it smells nutty. Slowly whisk in the milk until the mixture turns smooth and slightly thickened.
Stir the roux mixture directly into the Instant Pot with the broccoli and broth. Set the pot to sauté again and let it simmer gently for a couple of minutes until everything comes together.
Turn off the heat completely before adding the shredded cheddar, then stir it in a handful at a time. Adding cheese off direct heat keeps the soup smooth instead of turning grainy or oily.