Press the [Sauté] button and wait until the display says "hot". Then add olive oil and onions; stir occasionally for 2-3 minutes or until the onion is translucent. Add more olive oil, if needed, to prevent the onions from sticking to the pot.
Add garlic and cook until the garlic is fragrant, about 30 seconds.
Deglaze the pot by adding vegetable (or chicken) broth and loosen any bits stuck to the bottom of the pot to prevent the "burn" error message. Then, press the [Cancel] button.
Add broccoli, carrot, salt, pepper, paprika, and thyme. Place the lid back on and turn the knob to the "sealing" position. Press [Pressure Cook] (or Manual for older models) and set it to high pressure for 3 minutes. (It will take about 15 minutes for the Instant Pot to come to pressure)
While the instant pot is coming to pressure, in a small bowl melt butter in the microwave. If the half and half is coming straight from the fridge, microwave it for 15 seconds so that it becomes room temperature. Whisk together the butter, flour, and half and half. Then, set it aside for later.
When it is done cooking, quick release the pressure by turning the knob to "venting". Open the lid and give it a quick stir. Whisk in the butter/milk/flour mixture until it is fully mixed. Vigorously stir while adding the grated cheddar cheese. Do not let the cheese stick to the bottom of the pot.
Use an immersion blender to puree the soup to your desired consistency.
Serve while hot!