Warm the olive oil in a large pot over medium heat. Add the chopped onion and carrot and cook, stirring occasionally, until softened and translucent, 8–10 minutes.
Add the sliced garlic and cook until fragrant, about 1 minute.
Stir in the tomato paste and cook to caramelize it slightly, 1–2 minutes.
Add the canned whole peeled tomatoes with their juices (crush them by hand as they go in), the low-sodium vegetable broth, bay leaf, 1 tsp of the kosher salt, and the black pepper. Bring to a simmer, then reduce heat to medium-low and cook uncovered until the carrots are very tender and the flavors meld, 20–25 minutes.
Stir in about 1 cup of the basil (reserve the remaining 0.5 cup). Simmer 2 minutes, then remove from heat. Discard the bay leaf.
Blend the soup until smooth using an immersion blender. Alternatively, blend in batches in a countertop blender, venting the lid and covering with a towel; return soup to the pot.
Add the remaining basil and blend briefly for green flecks, or stir it in for a fresher texture. Taste and season with the remaining 0.25 tsp salt, if needed. Rewarm gently and serve hot.