Place the trivet or a steamer basket in the Instant Pot and add 1 cup of water. Arrange the trimmed Brussels sprouts on the trivet.
Close the Instant Pot, set the vent to "Sealing," and pressure cook on high for 2-3 minutes. Perform a quick pressure release when done.
Remove the Brussels sprouts from the Instant Pot and set them aside.
In the Instant Pot, combine the grated cheddar cheese, heavy cream, vegetable broth, minced garlic, dried thyme, paprika, salt, and pepper. Select the sauté function and cook, stirring, until the mixture is well combined and slightly thickened.
Add the steamed Brussels sprouts and mushrooms to the cheese mixture and stir to coat.
Transfer the mixture to a heatproof dish, sprinkle with grated Parmesan cheese (if using), and broil in the oven for a few minutes until the top is golden brown.