Preheat the oven to 400°F.
Place beets in a small roasting pan with the thyme, orange zest peel, and garlic cloves. Season with a ½ teaspoon of kosher salt and ¼ teaspoon of black pepper, then toss everything in 3 tablespoons of olive oil. Cover the pan with foil, sealing the edges, and roast for 40 minutes.
Halfway through the roasting, bring a large pot of water to boil and season with ¼ cup kosher salt when boiling. Add the pasta and cook until al dente, following the instructions on the package. Reserve 1 cup of pasta water then drain the pasta in a colander and set aside. Keep the pot on the stove.
To a blender jar, add the roasted beets, garlic, thyme leaves (discarding the stems and orange zest peel), 5 ounces of feta, orange juice, ¼ cup olive oil, ½ cup reserved pasta water, and red wine vinegar. Blend on high for about a minute or until creamy and smooth.
To the pot you cooked the pasta in, add the pasta back in and pour the beet sauce all over. Toss to coat evenly. You can warm it on low if the pasta cooled down and add more pasta water to thin if needed.
To plate, add the pasta first then crumbled on some of the remaining 3 ounces of feta and sprinkle on some fresh dill. Finish with some black pepper and a drizzle of olive oil.