Combine swiss chard and 1/4 cup water in a large dutch oven; tightly packed leaves and cover. Cook over medium heat until volume is reduced by half. Drain in colander, reserving 1/2 cup cooking liquid.
Heat oil over medium high heat and brown sausage - turning frequently - about 4 minutes or until browned. Add garlic and cook 30 seconds. Stir in beans and tomatoes with juice, breaking up the tomatoes. Bring to a boil and cook 3 minutes.
Reduce heat to medium-low. Stir in swiss chard with reserved cooking liquid (add additional cooking liquid if needed), potatoes, and red pepper. Cover and cook 20 minutes, until potatoes are tender. Sir cheese into minestra; remove from heat and let stand until cheese is partially melted, about 5 minutes. Ladle into bowls.