Chop the carrot, onion, and garlic.
Heat the oil in a soup pot over medium heat until it shimmers. Add in the onion, garlic and carrot and cook until the veggies until very soft and lightly browned, about 7-10 minutes.
While the veggies are cooking, dice the tomato, cut the corn from the cobs, and snip the green beans into 1 inch pieces. If you plan to add any other vegetables, now is the time to cut them up. If you are using canned or frozen veggies, get them ready to add.
Increase heat to high, stir in stock, tomatoes, bay leaves, thyme, rosemary and parmesan rind. Bring to a boil, then reduce the heat to a light simmer, stir in the green beans and cannelini beans and cook until soft and tender, about 10 minutes longer. Add corn to soup, simmer just another minute. Season to taste with salt and pepper. Serve hot.