In a large bowl, mix together the turkey, carrots, green beans, celery, green pepper, Cheddar cheese, garlic, dill, salt, and mayonnaise until thoroughly combined. Spread the filling out into a 9x12-inch baking dish.
In a separate bowl, cut the flour and shortening together with a pastry cutter until the mixture looks like crumbs; stir in water, 1 tablespoon at a time, just until the dough holds together. On a floured surface, roll the dough out to a rectangle slightly larger than the baking dish. Place the dough over the filling so that the edge of the crust fits over the edge of the dish. Cut several slits in the dough with a sharp knife; cover the dish with foil.
Bake in the preheated oven until the crust is browned and the filling is bubbly, about 40 minutes.