Using a mandolin, food processor, or sharp knife, cut all vegetables into approximately 1/8 inch slices.
In a separate bowl, combine Asiago and Provolone cheese together.
In a medium sized saucepan, combine half & half, 3 thyme sprigs, and garlic cloves. Bring to a simmer and turn off burner. Allow to steep for at least 15 minutes. Strain out solids and set aside.
In a buttered 13x9 inch-baking pan, layer first one layer of potatoes, one layer of sweet potatoes, and one layer of celery root. NOTE: between each layer, sprinkle with salt, pepper, and approximately ¼ cup of the cheese mixture. This recipe should yield two layers of each vegetable.
Gently pour reserved half & half over gratin.
Top with approximately 2 tablespoons fresh thyme. Sprinkle with remaining cheese and top with 3 whole sprigs of thyme (optional).
Place baking dish on a cookie sheet to catch any boil over. Bake in a 375-degree oven for at least ½ hours or until a sharp knife inserted into the middle of the pan shows your vegetables are soft.
Let cool for about 20-30 minutes before cutting and serving.
In medium saucepan, cook carrots in carrot juice until carrots are tender. Using a blender and immersion blender, puree until mixture is smooth. Stir in butter and continue blending briefly until creamy and satiny.