Turn Instant Pot to Sauté (Normal). Add butter and oil, then black cardamom seeds, peppercorns, cumin seeds; sauté until aromatic.
Add onion & garlic. Sauté for 3–4 minutes until the onions turn soft and lightly golden.
Stir in tomatoes, beetroot, carrot, kashmiri red chili powder, salt & sugar.
Add water/broth (at least 1 cup so the pot can come to pressure), deglaze well, and cancel Sauté.
Close the lid, set valve to Sealing, and you can either Pressure Cook on High for 5 minutes if you are in a rush or use the soup mode and set it to 15 minutes. Let pressure release naturally for 10 minutes, then quick release the rest.
Open the lid, allow the ingredients to cool and blend everything to a smooth purée with an immersion blender (or in batches in a regular blender).
If you want a silky restaurant-style texture, strain the purée. Adjust thickness with more water/stock if needed (although I didn't need to), and turn Sauté on Low.
Add sugar, ketchup, and black pepper. Let it simmer 2–3 minutes, tasting and balancing tang/sweet/salt.
Turn off Sauté, then finish with cream and a small knob of butter; stir gently.
Serve hot with lots of croutons.