Preheat the oven to 375 degrees F. Wash and peel beets. Cut each beet into eight wedges. Place in a 13x9x2-inch baking pan. Toss with the olive oil, salt, and 1/4 teaspoon pepper. Cover the pan tightly with foil. Roast for 20 minutes. Remove foil from the pan; stir beets gently. Roast, uncovered, for 20 to 25 minutes more or until tender. Set six wedges aside as a garnish.
Meanwhile, in a large saucepan cook the onion in hot butter until tender. Add the garlic; cook and stir for 30 seconds. Add rice; cook and stir for 2 minutes more. Remove from heat. Stir in the Prosecco. Add the chicken broth, thyme sprigs, and 1/4 teaspoon pepper. Bring to boiling; reduce heat. Simmer, covered, for 20 minutes (do not lift lid). Remove from heat. Let stand, covered, for 5 minutes. (Rice should be tender but slightly firm and mixture should be creamy.)
Remove and discard thyme sprigs. Stir in the 18 beet wedges and the 1/4 cup Asiago cheese. Transfer mixture to a serving bowl. Garnish with reserved beet wedges, thyme leaves, and Asiago shavings.