Preheat the oven to 375 degrees F. Heat the olive oil in a large skillet over medium heat. Add the onion, celery, carrot, garlic and 1/4 teaspoon each salt and pepper; cook, stirring occasionally, until the vegetables are soft and golden, about 7 minutes.
Add the chard, turkey and 1/4 cup water to the skillet; cook, stirring, until the chard wilts slightly, about 3 minutes. Add the tomatoes with their juice, increase the heat to medium high and simmer until slightly reduced, about 5 minutes. Add the pinto beans, then add the navy beans and their liquid. Add the parsley, thyme and oregano and return to a simmer.
Coarsely mash about one-quarter of the beans in the skillet with a potato masher or fork to thicken the mixture; season with salt. Transfer to a 2-quart baking dish. Cover and bake 45 minutes, then uncover and bake 10 more minutes.