Serve with Poppy-Seed Lavash, or substitute with store-bought lavash, lightly brushed with extra-virgin olive oil, sprinkled with poppy seeds, and warmed in the oven.
Bring a large pot of water to a boil and generously season with coarse salt. Add all beets, reduce heat, and simmer until beets are tender and easily pierced with a knife, about 15 minutes. Drain beets; when cool enough to handle, gently rub with a paper towel (or use a paring knife) to remove the skin, keeping as much of the root end intact as possible. Transfer beets to a bowl, cover to keep warm, and set aside.
Melt butter in a medium saute pan over medium heat. Add allspice, bay leaves, and cinnamon and cook until butter is nutty in aroma and golden brown, about 5 minutes. Pour butter through a fine sieve into a bowl; discard solids. Whisk 1 teaspoon coarse salt and hazelnuts into brown butter.
Spread yogurt on a platter and sprinkle with flaky salt. Arrange beets on yogurt, spoon brown butter and nuts on top, and sprinkle with dill and more flaky salt. Garnish with rose petals and drizzle with pickling liquid; serve immediately.