In a bowl, mix 1 cup flour and 1/4 teaspoon salt. Add 4 tablespoons of the butter and all the shortening. With a pastry blender or your fingers, cut or rub in butter and shortening until mixture forms pea-size pieces.
Sprinkle 2 tablespoons cold water over mixture and stir with a fork just until evenly moistened. With lightly floured hands, gently squeeze about 1/4 cup of the dough into a ball. If it won't hold together, crumble lump back into bowl and sprinkle with 1 more tablespoon water; stir again with a fork until evenly moistened.
With lightly floured hands, gather all of the dough into a ball. Pat into a 6-inch disk, wrap in plastic, and chill until firm, 20 to 30 minutes.
Meanwhile, peel beets, carrots, turnips, and onions; trim off and discard tops and ends. Cut turnips and beets into about 1-inch chunks. Peel garlic, but keep cloves whole. You should have about 8 cups vegetables total; save any extra for other uses.
In a 10- to 11-inch ovenproof frying pan (with 2- to 2 3/4-qt. capacity), melt remaining 1 tablespoon butter with oil over medium heat. Add 1/2 cup broth, beets, carrots, turnips, onions, and garlic. Bring to a boil, then reduce heat, cover, and simmer, stirring occasionally, until vegetables are tender when pierced, 20 to 30 minutes. Uncover and add remaining 1/4 cup broth and the vinegar, chopped herbs, and sugar. Stirring often to prevent scorching, cook over high heat until most of the liquid is evaporated and pan juices are thick and syrupy, 3 to 5 minutes. Remove pan from heat and season vegetables to taste with salt and pepper. Set aside.
On a board, overlap sheets of waxed paper to make a 16-inch square. Unwrap dough, dust with flour, and place on paper. Cover with more waxed paper. With short, gentle strokes, roll dough into an even 12-inch round. Remove top layer of waxed paper, invert dough over vegetables, and remove remaining paper. Tuck edge of dough between vegetables and pan. Cut four 1-inch vents near center of dough.
Bake in the middle of a 400° regular or convection oven until pastry is deep golden, 30 to 40 minutes. Let cool on a rack for 5 minutes. Loosen crust from sides of pan. Place a rimmed serving plate over crust. Using pot holders, hold plate and pan together, invert pan over plate, and lift pan off. Garnish tart with fresh herb sprigs. Cut into wedges and serve with sour cream to spoon over individual portions.