Preheat the oven to 425°F (220°C).
In a medium bowl, combine the chicken pieces, basil pesto, chopped shallot, lemon zest, and a pinch of salt, black pepper, and crushed red pepper flakes; let sit 15 minutes to marinate.
Toss the potato pieces with 2 tablespoons olive oil, salt, and black pepper on a baking sheet; spread in a single layer and roast for 20 minutes, until tender.
Remove the sheet from the oven; add the butter, lemon juice (from the zested lemons), garlic, and sesame seeds to the potatoes and toss to coat. Push the potatoes to one side of the pan.
Arrange the marinated chicken on the empty side of the baking sheet and return to the oven; roast for 20 more minutes, until the chicken is cooked through and potatoes are golden.
Meanwhile, make the sun-dried tomato vinaigrette: whisk together 1/4 cup of the reserved sun-dried tomato oil, 2 tablespoons lemon juice, and 2 tablespoons balsamic vinegar; stir in the chopped sun-dried tomatoes and chopped dill or thyme, and season to taste with salt, pepper, and red pepper flakes.
Make the whipped feta: combine the feta, 1/4 cup Greek yogurt, and 2 tablespoons olive oil in a food processor and pulse until smooth and creamy.
To serve, spread whipped feta on plates, drizzle with the sun-dried tomato vinaigrette, add the roasted potatoes and pesto chicken, and serve with pita or naan and fresh herbs.