Peel the potato and cut into 1/2 inch pieces. In a medium saucepan, cover the potato with room-temperature salted water. Bring the potato to a boil, then reduce the heat to a simmer and cook until tender, about 20 minutes. Drain the potato, then mash together with the milk, butter and salt and pepper to taste. Let the mashed potato cool to warm.
Place each chicken breast, smooth-side-up, on a work surface. Place the palm of one hand on top of one breast, then insert a pairing knife to make a length-wise pocket in each breast. Divide the mashed potato filling between the breasts, stuffing them full. Rub the chicken with the oil, then season all over with 3/4 teaspoon salt and 1/2 tea-spoon pepper.
Preheat the oven to 425°F.
Place the chicken breasts on an oiled baking sheet and bake until cooked through, about 30 minutes. Let stand 5 minutes before slicing and serving.
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