Heat the olive oil in a Dutch oven or large, heavy bottomed pot. Add the onion, carrots, and celery and cook until beginning to soften, about 4-5 minutes. Sprinkle with salt and pepper.
Add the Italian sausage, breaking it up into smaller pieces. Cook, stirring occasionally, until browned. Stir in the garlic cloves and saute until fragrant, about 1 minute. If there's a lot of fat in the pan, drain or use paper towels to sop up all but about 1 Tablespoon of fat.
Pour in the chicken broth and beans and bring the mixture to a boil. Reduce heat to low, and add the rainbow chard, stirring until wilted, about 3 minutes. Taste and season as needed.
Serve with Parmesan flakes and crusty bread.