How to Make It
Preheat oven to 450°F. Remove tops from carrots, reserving 1/4 cup. (Discard remaining tops or reserve for another use.) Cut any large carrots in half so they are all roughly the same size. Toss together carrots, olive oil, salt, and pepper in a large bowl. Spread in an even layer on a rimmed baking sheet. Roast in preheated oven until carrots are browned in spots and tender, about 20 minutes, stirring once halfway through cooking.
While carrots roast, melt butter in a medium saucepan over medium. Cook, stirring often, until butter browns and smells nutty, about 5 minutes. Remove from heat, and let cool 1 minute. Carefully stir in hazelnuts, sherry vinegar, lemon juice, and salt and pepper to taste.
Spread red beet puree in center of a large platter; arrange carrots on top. Spoon hazelnut sauce over carrots. Tuck dandelion greens and reserved carrot tops around carrots. Using a vegetable peeler, shave cheese over carrots.