Preheat the oven to 400°F.
Pouro 2 tablespoons of the olive oil directly into the can of tomatoes. Add 1/2 teaspoon of the salt, 1/4 teaspoon of the pepper, and the oregano and stir to combine.
Strip the chard leaves off the stems. Thinly slice the stems and chop the leaves, keeping them in separate piles.
Pour the remaining 2 tablespoons of the olive oil into a large skillet over medium heat. When it shimmers, add the chard stems, onion, and garlic and cook, stirring frequently, until tender, about 8 minutes. Stir in the remaining 1/2 teaspoon salt, the remaining 1/4 teaspoon pepper, and the red pepper flakes and cook for 30 seconds. Stir in the chard leaves and cook, stirring, until they are tender, about 2 minutes. Turn off the heat, stir in the beans, taste, and and add more salt if needed. Use the back of a wooden spoon, a fork, or potato masher to lightly mash some of the beans.
Transfer the mixture to a bowl and let it cool slightly.
Lightly grease a 9x13-inch baking dish with olive oil. Spread 1 cup of the tomato sauce on the bottom of the dish.
Use a teaspoon (a long-handled ice tea spoon with a narrow bowl works particularly well) and your fingers to fill the dried pasta shells with the bean mixture, stuffing them as full as possible. Place the shells in the baking dish as you fill them, packing them tightly.
Pour the remaining tomato sauce over the shells, spreading it so it evenly covers the shells. Sprinkle on the mozzarella and Parmesan, cover the dish with aluminum foil, and bake until the sauce is bubbly and the pasta is tender, about 30 minutes. Uncover and bake until the cheese is lightly browned, about 10 minutes. Serve hot.
Note: If you cook dried cannellini beans in an InstaPot: Soaked overnight 5-8 minutes high pressure, natural release. UNsoaked 20-30 minutes high pressure, natural release.