In a medium bowl or in the bowl of a food processor, blend together the butter, Thai dry chili paste and 1 tablespoon kosher salt until it is a smooth paste. Set aside.
Season the chicken on all sides with salt and pepper. Carefully lift the skin of the chicken breast and spread some of the compound butter (about 1/2 tablespoon) between the skin and meat of the breast. Replace the skin and spread more of the butter on the skin.
Toss the potato slices in a drizzle of olive oil and season with salt and pepper, and about 2 tablespoons of additional Thai dry chili paste.
Place the chicken in a baking dish. Lightly toss the prepared potatoes and nestle between the chicken pieces. Season lightly with salt. Bake for about 30 minutes or until a thermometer inserted into chicken registers 165°F. Serve immediately.