Set Instant Pot to Sauté mode. Add 2 tablespoons neutral oil to Instant Pot and heat insert until oil is hot and shimmery.
When oil is hot, add ½ cup chopped yellow onion, 1 cup chopped carrots, and 2 tablespoons minced fresh garlic. Stir to incorporate, then sauté vegetables, stirring occasionally, 5 to 10 minutes or until onions and carrots are very soft.
When vegetables are soft, add 1 tablespoon organic tomato paste and stir 2 to 3 minutes or until tomato paste begins to caramelize, developing slight brown color.
Once tomato paste has caramelized slightly, pour in 1 28-ounce can whole San Marzano tomatoes with liquid.
Add 2 cups vegetable broth, ½ teaspoon dried thyme, 1 large bay leaf, and 2 tablespoons coconut aminos. Stir to incorporate all ingredients, making sure to scrape bottom of Instant Pot insert with spoon to loosen any bits of vegetables or garlic that may have stuck to bottom of insert.
When ingredients are incorporated, secure lid on Instant Pot with vent in Sealing position. Set Instant Pot to Manual High Pressure mode and set cook time to 15 minutes. Instant Pot will pressurize and then 15-minute cook time will begin.
When 15-minute cook time ends, carefully Quick Release internal pressure. Use caution as any escaping steam will be very hot. Remove lid from Instant Pot and set aside.
Turn off Instant Pot and let soup cool slightly, then remove bay leaf from Instant Pot and discard. Add 1 14-ounce can unsweetened, full-fat coconut cream and sea salt to taste. Stir to incorporate.
Once coconut cream is completely incorporated, ladle soup into high-speed blender, working in batches based on size of blender. Purée soup until completely smooth, then return soup to Instant Pot. Repeat until soup reaches desired consistency.
When satisfied with consistency of soup, ladle soup into serving bowls. Season each bowl with plenty of freshly ground black pepper and chopped fresh basil to taste. Serve warm.