Remove and discard crust from bread, and pulse bread in food processor into soft, small crumbs. Line a sheet pan with parchment paper. Combine breadcrumbs and rosemary in a small bowl. Pour 3 tablespoons olive oil onto a plate or a shallow bowl. Coat each round of goat cheese in olive oil, and dredge in breadcrumb mixture. Arrange cheese on the prepared parchment-lined pan or on a large plate, and place in the refrigerator to chill until firm, at least 1 hour or overnight.
Heat oven to 425 degrees. Line a baking sheet with aluminum foil. Cut beets in half, and toss with 2 tablespoons oil, 1 teaspoon salt, and 1/4 teaspoon pepper. Arrange beets in one layer on aluminum foil, cut side down, and place rosemary sprig on top of the beets. Cover beets with another piece of aluminum foil, and seal the edges all around, creating a rectangular packet. Bake on the lowest shelf of the oven until beets are fork tender, about 30 minutes. Let stand until cool enough to handle. Peel beets; cut the larger ones in half, and set aside.
Strip chard leaves from stems. Discard stems, or saute for later use. Rinse and drain the leaves. Do not dry them. Place chard leaves in a large pot over medium heat, and sprinkle them with 2 teaspoons salt. Cover pot, and cook over medium heat, opening the lid only to stir, until just wilted. Remove from the heat, return to the colander, and rinse with cold water to stop them from cooking. Using your hands, gently squeeze any excess water from chard, and coarsely chop; you should have about 2 cups. Set chard aside, but leave the pot on stove.
Meanwhile, fill a pasta pot with water, and set over high heat. Bring to a boil, add 1 tablespoon salt, and stir in pasta. Let cook until pasta is al dente, 7 to 10 minutes. Remove from heat, drain in a colander, and set aside.
Heat 3 tablespoons olive oil in the chard pot over medium heat. Add garlic, and cook slowly until garlic is just toasted, stirring often. Add chard, remaining 1 teaspoon rosemary, remaining 1/2 teaspoon salt, and remaining 1/2 teaspoon pepper, and saute until chard is wilted, about 3 minutes. Add pasta, toss, and cook just until hot. Adjust seasoning to taste with olive oil, salt, and pepper, and transfer pasta to a large serving platter. Arrange beets over pasta.
Bake cheese in the oven until very soft to the touch, heated through, and golden brown, 6 to 7 minutes. Remove from oven, arrange hot cheese around platter of pasta, and serve immediately.