Add enough of the oil to coat the bottom of a cast-iron skillet or heavy-bottomed saute pan over medium-high heat. Add the turkey in batches and lightly brown, adding more oil as necessary.
Divide the turkey into 4 even portions in the skillet. Place 2 slices of the Cheddar over each turkey portion. Open the buns and place over the cheese. Let sit until the buns warm up. Place a flat spatula underneath the turkey and flip the sandwich over. Top with the carrots and peppers. Serve to kiddies and make sure they take at least one monster bite of the sandwich with the veggies.