Cut the potatoes into 1/2-inch cubes, place in a large pot and cover with 1 inch of water. Salt the water and bring to a boil over high heat. Continue boiling until the potatoes are tender, about 10 minutes.
In a medium mixing bowl, combine the ground turkey with the breadcrumbs, egg, sage and 1/2 teaspoon salt. Divide the mixture into eighths and form into 1/2-inch thick patties.
Heat 1 tablespoon of EVOO in a large grill pan over medium heat. When the EVOO is rippling hot, add the patties and cook, flipping once, until cooked through, about 4 minutes per side. Place two small cheese squares on each burger.
When the potatoes are cooked, drain them and immediately toss with the remaining 1/4 cup EVOO, vinegar, 1/2 teaspoon salt and the chives.
Serve each burger on a roll spread with mustard. Serve two burgers on each plate with the potato salad.