Cover the potatoes and garlic with cold water. Heat over high heat until boiling, then cover and reduce heat to medium. Cook for 20 minutes, until the potato is cooked through.
Meanwhile, heat the olive oil in a deep frying pan over medium-high heat. Sear the chicken breasts until browned on both sides, about 2 minutes per side. Remove to a plate. Add the onion and carrots, as well as a bit of chicken broth if necessary to keep from sticking, and sauté until soft, 5-10 minutes.
Add the chicken back to the pan along with the broth. Cook until the chicken is cooked through, 15-20 minutes.
Remove the chicken and shred with two forks. Add back to the pan along with the peas, flour, pepper and herbes de Provence. Stir and cook 2-3 minutes, until thickened.
When the potatoes and garlic are cooked through, drain and return to the pot. Mash over low heat to remove excess moisture, then mash in the milk and butter, as well as salt to taste.
Pour the filling into a baking dish and cover with the mashed potato. Broil until the potato is browned on top.