Heat 2 tablespoons oil in a pressure cooker on the 'Sear' setting. Add onions; cook and stir until golden brown, 10 to 15 minutes.
Transfer onions to a food processor; grind into a paste. Remove to a bowl.
Combine tomatoes, garlic paste and ginger paste in the food processor; puree until smooth.
Heat remaining 2 tablespoons oil in the pressure cooker using the 'Sear' setting. Add onion puree; cook, stirring constantly, until browned, about 2 minutes. Stir in the tomato mixture. Add goat meat, water, potato, carrot, garam masala, coriander, cumin, turmeric, salt and cayenne pepper.
Close pressure cooker and seal according to manufacturer's instructions. Set the timer for 50 minutes at high pressure. Release pressure using the natural-release method according to manufacturer's instructions.