Select the Sauté setting on the Instant Pot and melt the butter. Add the onion, carrot, celery, garlic, and 1⁄2 teaspoon salt and sauté for about 5 minutes, until the onion is softened but not browned. Add the broccoli and vegetable broth.
Secure the lid and set the Pressure Release to Sealing. Press the Cancel button to reset the cooking program, then select the Pressure Cook or Manual setting and set the cooking time for 1 minute at high pressure. (The pot will take about 10 minutes to come up to pressure before the cooking program begins.)
While the soup is cooking, in a liquid measuring cup, stir together the half-and-half, cornstarch, and hot sauce.
When the cooking program ends, let the pressure release naturally for 5 minutes, then move the Pressure Release to Venting to release any remaining steam. Open the pot, press the Cancel button to reset the cooking program, then select the Sauté setting. Stir the half-and-half mixture to recombine, then pour it into the pot. Stir in the cheese. Let the soup come up to a simmer and cook for 1 to 2 minutes, stirring occasionally, until it has thickened slightly. Press the Cancel button to turn off the Instant Pot. Taste and adjust the seasoning with salt and hot sauce if needed.
Ladle the soup into bowls and serve piping hot.