Turn Instant Pot on to Sauté and select Normal. Let the pot warm up for a few minutes.
Add olive oil, chopped onions, garlic, salt, pepper, thyme, oregano, and red pepper flakes to the Instant Pot and stir. Sauté until onions become translucent; about 3–5 minutes. Turn the Instant Pot off.
Add tomato paste, canned tomatoes, and vegetable broth to the Instant Pot and fasten the cover on securely. Make sure the steam release handle is on correctly. Turn the Instant Pot on to Pressure Cook and select More and High. Set to 4 minutes.
Once the Instant Pot timer goes off, let the pressure out by turning the steam release handle. The float valve will drop when the pot is depressurized. Remove the lid.
Add the basil leaves. Using an immersion blender, blend the contents of the pot until the tomatoes have become completely smooth.
Add goat cheese, grated Parmesan and cream while the soup is still hot. Stir until cheese has melted and soup has an even, smooth consistency. If you need to use the immersion blender again, do so for only a few pulses.
Pour into individual bowls and top with more Parmesan cheese and some basil leaves. Enjoy!