Preheat oven to 425° F. In a medium bowl, combine the chicken, pesto, shallot, lemon zest, and a pinch each of chili flakes, salt, and pepper. Let sit 15 minutes.On a baking sheet, toss the potatoes with olive oil. Season with salt and black pepper. Roast for 20 minutes, or until tender. Remove the potatoes from the oven. Add the butter, lemon juice, garlic, and sesame seeds, toss to coat, then push the potatoes to one side of the pan. Put the chicken on the other side. Return the pan to the oven and roast for 20 minutes more, until the chicken is cooked through.Meanwhile, make the tomato vinaigrette. In a bowl, whisk together 1/4 cup of the reserved sun-dried tomato oil, 2 tablespoons lemon juice, and the balsamic vinegar until combined. Stir in the sun-dried tomatoes, dill, and season with salt, pepper, and red pepper flakes.To make the feta. Combine all ingredients in a food processor and pulse until smooth and creamy. The feta can be kept in the fridge for up to 3 days. To serve, spread the feta onto plates and drizzle with the vinaigrette. Add the potatoes and chicken, top with fresh herbs. Serve pita on the side. Enjoy!