Preheat oven to 375 degrees. On a lightly floured surface, roll out dough 1/8 inch thick. Press firmly into a removable bottom tart pan; press firmly against pan. Pierce bottom of shell all over with a fork. Refrigerate or freeze until firm, about 30 minutes.
Line shell with parchment; fill with pie weights or dried beans. Bake until golden brown, about 30 minutes. Remove weights and parchment. Transfer to a wire rack to cool completely. (Keep oven on.)
Trim all but 1/2 inch of stems from beets; rinse well. Wrap each beet individually in aluminum foil. Transfer to a rimmed baking sheet. Roast until beets are tender, 45 minutes to 1 hour. When cool enough to handle, peel beets by simply pinching the skin; the skin will come off with ease. Cut into thin rounds. Raise heat to 425 degrees.
Stir together goat cheese, ricotta, and chopped basil & parsley until well combined; season with pepper. Spread mixture over tart shell, filling all the way to edges.
Arrange beets over cheese mixture, overlapping slices slightly and alternating colors, if possible. Lightly season with salt. Sprinkle with thyme leaves on top. Lightly drizzle with oil, and season with pepper. Bake until golden brown, about 25 minutes. Serve warm.