Preheat the oven to 425 degrees. On a sheet pan, toss the potatoes with 2 tablespoons of olive oil and season generously with salt and pepper. Roast for 20-25 minutes, tossing halfway through.
Meanwhile, heat a large oven-safe skillet over medium-high heat with 2 tablespoons of olive oil. Once hot, sear the chicken about 2-3 minutes on each side until golden brown. During the last couple minutes, add the tomatoes, fresh thyme, garlic, and last 2 tablespoons of olive oil, then season lightly with salt, pepper and a pinch of red pepper flakes.
Transfer the skillet to the oven and finish cooking alongside the potatoes, about 8-12 minutes, or until the chicken is cooked through. Remove from the oven and gently smash some of the tomatoes with a fork to release their juices.
To make the whipped feta, add all the ingredients to a food processor or blender. Pulse until smooth and creamy.
To serve, spread the feta onto plates and top with potatoes and chicken, then the tomato sauce. Garnish with fresh basil.