For the chicken kebab filling, stir together all of the ingredients for the marinade. Cube the chicken breasts, stir them into the marinade, transfer everything to the refrigerator, and chill for at least 4 hours, preferably overnight.
Preheat the oven to 350°F. For the phyllo dough pockets, stack 3 sheets of phyllo dough on top of each other, then place them on top of a 4 x 4-inch casserole dish.
Briefly sauté the vegetables in a frying pan with a little bit of olive oil and season them with salt & pepper. Spoon the vegetables into the center of the phyllo dough, filling the small casserole dish up halfway.
Briefly sear the marinated chicken in a frying pan along with the excess marinade, then fill the small casserole dish with the chicken kebab filling.
Fold the phyllo dough over the filling to make a small, sealed pocket. Repeat this process until you've made 12 phyllo dough pockets.
Brush melted butter over the surface of a 12 x 8-inch casserole dish and place 6 phyllo dough pockets in it. For the mint pesto, put all of the ingredients into a food processor or blender and blend them until a smooth, creamy sauce forms.
Brush some melted butter over the phyllo dough pockets, then spoon 1 tablespoon of mint pesto on top of each one. Spread the pesto evenly over all 6 phyllo dough pockets and place 6 slices of cheese on top of the layer of pesto.
Place the remaining 6 phyllo dough pockets on top of the layer of cheese, coat them in melted butter, and place another slice of cheese on top of each phyllo dough pocket. Transfer the casserole dish to the oven and bake the chicken kebab phyllo dough pockets at 350°F for 30 minutes, until golden brown.