Preheat oven to 375°. Trim beet stems, if attached, leaving about 1/4 in. Put red and golden beets in separate baking dishes. To each, add 2 tbsp. vinegar, 3 allspice berries, 1 garlic clove, lightly crushed, and 1/4 in. boiling water. Cover tightly with foil and bake until beets are tender when pierced, about 1 hour. Let cool. Peel beets, slice into slim wedges, and put in separate bowls.
Grate or mince remaining 2 garlic cloves (Fletcher recommends using a Microplane.) Stir half into yogurt cheese with 1/2 tsp. salt and about 2 tsp. oil.
In a small bowl, whisk together remaining 2 tsp. vinegar, garlic, 1/4 tsp. salt, and 2 tbsp. oil; set dressing aside. Toast fennel seeds in a small frying pan over medium heat, swirling often, until a shade darker, 4 to 5 minutes. Let cool, then pound in a mortar until fine. Add to dressing.
Dress red and golden beets separately with vinaigrette, adding just enough to coat them lightly. (You may not need it all.) Add more salt to taste if you like.
With a rubber spatula, spread yogurt cheese about 1/2 in. thick on 1 side of a platter, creating some indentations where oil can pool. Sprinkle with feta, then drizzle with 2 tbsp. oil and sprinkle with pumpkin seeds. Arrange beets next to yogurt cheese and spoon a little more dressing over them. Sprinkle everything with zaatar and red pepper. Serve with pita bread.
* Find at well-stocked grocery stores and worldspice.com.