Cut the potatoes into 1/2-inch cubes, place in a large pot and cover with an 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 bread crumbs, egg, sage and 1/2 teaspoon of salt. Divide the mixture into eighths and form into 1/2-inch-thick patties.
Heat the 1 tablespoon of olive oil in a large grill pan over medium heat. When the oil is rippling hot, add the patties and cook, flipping once, until cooked through, about 4 minutes per side. Place 2 small cheese squares on each burger.
When the potatoes are cooked, drain them and immediately toss with the remaining 1/4 cup of olive oil, the vinegar, 1/2 teaspoon of salt and the chives.
Serve each burger on a roll spread with mustard. Serve 2 burgers on each plate with the potato salad.