Gather all your ingredients: dice the carrots, celery, and onion into small, even pieces for quick, uniform cooking.
Heat 2 tablespoons of olive oil in a skillet over medium heat until it shimmers and starts to smell fragrant.
Add the diced onions, carrots, and celery to the skillet. Sauté for about 5-7 minutes until the vegetables are fragrant, softened, and slightly golden around the edges.
Transfer the sautéed vegetables into your slow cooker, spreading them evenly at the bottom.
Pour in the crushed tomatoes and vegetable broth, then add the drained white beans, dried thyme, bay leaf, and a pinch of salt and pepper. Stir everything together.
Cover the slow cooker with the lid and cook on low for 6 to 8 hours, allowing the flavors to meld and the vegetables to become tender.
About 30 minutes before serving, stir in the small pasta and chopped greens. Cover and continue cooking until the pasta is al dente and greens are wilted, approximately 30 minutes.
Remove the bay leaf and taste the soup, adjusting salt and pepper as needed. Drizzle with 2 tablespoons of olive oil for a velvety finish.
Serve hot, garnished with fresh basil leaves for a bright, fragrant touch. Enjoy your hearty, comforting minestrone!