Trim the turkey as needed, cut lengthwise, through each one, stopping short of the other side, and open like a book.
Season lightly with salt and pepper. Rub each with 1 tablespoon pesto. Divide cheese pieces evenly.
Shut tenders, close with toothpicks.
Season outside lightly. Spread evenly with remaining pesto.
Place on a sheet, or on a large rack (like a cookie rack, set over a jelly sheet), and bake in the middle or upper-middle level of a preheated 375 degree oven till done, approximately 20 minutes. Rest two or three minutes before slicing, and serving at once.