Start by chopping the carrots, celery, and onion into small, even dice. Mince the garlic and set everything aside for easy access during cooking.
Pour the olive oil into your large slow cooker and turn it to low heat. Add the diced onions, carrots, and celery, stirring to coat them with the oil. Let them cook for about 5 minutes until they start to soften and become fragrant.
Add the minced garlic, dried oregano, and thyme to the slow cooker. Stir everything together and cook for another minute until the garlic is fragrant and slightly translucent.
Pour in the canned diced tomatoes (with their juices) along with the drained cannellini beans. Give everything a good stir to combine the flavors.
Add the vegetable broth and the small pasta to the slow cooker. Stir gently to distribute everything evenly, ensuring the pasta is submerged in the liquid.
Cover the slow cooker and set it to low. Let everything simmer for 6 to 8 hours, or until the vegetables are tender and the flavors are well melded. The kitchen will fill with the aroma of herbs, garlic, and simmering tomato.
About 30 minutes before serving, stir in the chopped spinach or kale. Cover again and let it wilt into the hot soup, adding a vibrant green touch and fresh flavor.
Taste the soup and adjust the seasoning with salt and pepper. If desired, add a splash of lemon juice or a drizzle of olive oil for brightness.
Use a ladle to serve the minestrone hot into bowls, making sure to include plenty of vegetables, beans, and pasta. Garnish with grated Parmesan if you like, and enjoy the cozy, layered flavors.