Heat the butter and olive oil in the Instant Pot using the saute function. Once the pot is hot, add the onion and saute for 3 minutes, stirring often.
Add the garlic and cook, stirring, for 30 seconds. Turn off the Instant Pot.
Pour in the 3 ½ cups of broth. Stir and scrape up any bits that are stuck to the bottom of the pot.
Stir in the Italian seasoning, salt and pepper. Then stir in the potatoes, carrots and broccoli.
Close the Instant Pot lid and move the steam release valve to the sealing position. Pressure cook on high pressure for 5 minutes. The Instant Pot will take about 15 minutes to reach pressure and then the cook time will start counting down.
When the cook time ends, allow the pot to release naturally for 10 minutes. Then carefully move the steam release valve to the venting position and wait for all of the remaining steam to escape. Once the pin drops down, open the Instant Pot lid.
Use a slotted spoon to remove about 1 cup's worth of broccoli pieces and set aside, coarsely chop into bite-size pieces if large.
Use an immersion blender to puree the soup in the pot until it is mostly smooth. (If you do not have an immersion blender, you can transfer the soup in batches to a countertop blender.)
Stir in the shredded cheddar cheese until melted. Taste and add more salt if needed, to taste. Stir in the reserved broccoli pieces. Serve.