Heat the olive oil in a large, flat frying pan over medium-high heat. Once shimmering, add the chicken thighs, skin-side down, and sear until the skin is deep golden brown and crisp. Flip the chicken and sear the other side for just 1 to 2 minutes, then remove the chicken to a plate and set aside.
Add the pre-cooked carrots and potatoes to the same pan, sprinkling them with a little more Cajun seasoning and pepper. Sauté them for a few minutes until they start to brown and become aromatic, picking up all those browned bits from the chicken. Remove the vegetables from the pan.
Using a small knife, poke several shallow slits into the skin of each chicken thigh. Stuff these openings with the garlic cloves and fresh rosemary sprigs. Arrange the chicken thighs, carrots, and potatoes in a baking tray or oven-safe pan, then pour any remaining olive oil from the frying pan over everything.