Preheat oven to 450°F. Pat chicken dry; season generously with salt and pepper.
Heat a large, heavy ovenproof skillet (such as cast iron) over medium-high. Add butter and 1 tablespoon oil.
When butter melts and foam subsides, add chicken, skin-sides down. Cook, undisturbed, until skin lifts easily from skillet and forms a golden-brown crust, about 5 minutes. Transfer to a plate, skin-sides up.
Add leeks, carrots, bread cubes, garlic, lemon zest, and remaining 2 tablespoons oil to skillet, stirring to evenly coat; season with salt and pepper.
Return chicken and any accumulated juices to skillet, arranging breasts on top of vegetables in a single layer, skin-sides up.
Roast until a thermometer inserted into thickest part of breasts reads 160°F, 25 to 30 minutes.
Transfer chicken to a plate and loosely cover to keep warm.
Stir mint into skillet; return to oven. Roast until vegetables are tender and golden and bread turns crisp, 10 to 12 minutes.
Stir 1 ½ tablespoons lemon juice into vegetable mixture and sprinkle with more mint leaves.
Drizzle remaining 1 ½ tablespoons lemon juice over rested chicken. Carve chicken; serve with vegetable mixture.
