Preheat oven to 375 degrees. Heat a large skillet over medium-high heat. Add 2 tablespoons oil, onion, and chard stems. Cook, stirring occasionally, until soft, about 4 minutes. Stir in chard leaves, season with salt, and cook until tender, about 2 minutes. Transfer to a plate. Wipe skillet; return to medium-high heat. Add remaining 1 tablespoon oil, garlic, and pepper flakes. Cook until fragrant, about 30 seconds. Add tomatoes with juices and simmer, breaking into pieces, until thickened, about 3 minutes. Season with salt.

Spread 1/2 cup tomato sauce in the bottom of an 8-inch square baking dish. Top with 2 lasagna noodles, 3/4 cup sauce, one-third of chard mixture, and 1 cup cheese. Repeat layering twice. Top with remaining noodles, sauce, and cheese. Loosely cover with parchment-lined foil. Bake 30 minutes. Uncover and bake until bubbly, about 15 minutes. Let cool 20 minutes, then slice and serve.