Cook, stirring regularly, until the onion has softened, about 3 minutes.
Add the garlic and sauté 30 seconds, stirring the entire time.
Press cancel.
Add the broccoli, carrot, broth, salt, and pepper. Do not stir.
Close and lock the lid. Be sure that the pressure release valve is set to sealing.
Press manual high pressure and adjust the time to 1 minute.
When the cooking time has finished, do a controlled quick release.
Remove the lid, and give the soup a good stir.
In a small bowl whisk together ¼ cup cornstarch with ¼ cup broth until well-combined.
Press sauté and add the cornstarch mixture to the Instant Pot.
Cook, stirring constantly, the soup thickens.
Press cancel.
Add about ¼ cup of cheese and stir in until melted.
Repeat with the remaining cheese.
Once all of the cheese has been added and has melted, stir in the cream.
Serve topped with additional black pepper, if desired.
Notes
Onion: I recommend using a sweet onion for this recipe. If you don't want to use a sweet onion, white onion or yellow onion will work well.
Broccoli: It's important that the frozen broccoli be in small pieces. If you'd like to use fresh broccoli, cook for 2 minutes.
Broth: If you'd like to keep the soup vegetarian, use vegetable broth. Otherwise, chicken broth will also work.
Sharp cheddar cheese: Sharp cheddar cheese gives the soup stronger flavor. If you'd like milder cheese flavor, you can use mild cheddar cheese or a cheddar cheese blend.
Cheese: I recommend using freshly grated cheese because it should melt well. However, I've used bagged cheese, and I've never had any issues.
Cream: Half-and-half could also be used. It's a little thinner, so it will give you slightly thinner soup.