Heat 1 tablespoon oil in a medium Dutch oven or a large pot over medium heat. Cook onions and carrot until just softened, about 5 minutes. Season with 2 1/2 teaspoons salt. Stir in minced garlic, thyme, and red-pepper flakes. Cook for 1 minute.
Drain and rinse cannellini beans. Add to Dutch oven with water and bay leaf. Bring to a boil. Reduce heat to low. Simmer, covered, until beans are tender, about 45 minutes. Let cool slightly. Discard bay leaf.
Working in batches, puree soup in a blender until smooth, transferring it to a clean pot as you work. (For safety, remove cap from hole in lid, and cover with a dish towel to prevent spattering.)
Bring soup to a simmer. Stir in kale and 1/4 teaspoon pepper. Simmer until kale is tender, about 15 minutes.
Meanwhile, heat remaining 2 tablespoons oil in a small skillet over medium-low heat. Cook sliced garlic, stirring often, until crisp and golden, 1 to 2 minutes. Transfer garlic to a plate using a slotted spoon. Reserve garlic oil.
Divide soup among 6 bowls. Garnish with garlic chips, and drizzle with garlic oil.
Pureed cannellini beans give this soup a velvety texture, canceling out the need for heavy cream.