Preheat oven to 200°F. Toss together tomatoes and 2 teaspoons oil on a rimmed baking sheet lined with parchment paper. Arrange tomatoes cut sides up. Roast at 200°F until softened and dehydrated, about 5 hours. Set aside.
Remove stems from chard, and thinly slice stems to equal 1 cup. Discard remaining stems, or reserve for another use. Tear leaves into 1-inch pieces.
Bring a large pot of water to a boil over high. Add pasta; cook 4 minutes. Add sliced chard stems; cook until pasta is al dente and stems are tender, about 3 minutes, adding chard leaves during last 20 seconds. Reserve 1 cup cooking liquid. Drain pasta mixture.
Heat garlic and 1/4 cup oil in a large skillet over medium. Cook, stirring often, until garlic is light golden brown, about 4 minutes. Add pasta-chard mixture, roasted tomatoes, salt, crushed red pepper, and reserved cooking liquid. Cook, tossing constantly, until sauce thickens and pasta is coated, 3 to 4 minutes. Divide pasta among 4 shallow bowls. Top evenly with ricotta, pine nuts, and remaining 1 teaspoon oil.