Place the potatoes in a saucepan. Cover with cold water. Simmer until tender, 12 to 15 minutes. Drain, run under cold water, and cut into ½-inch pieces.
In a bowl, whisk the mustard, lemon juice, oil, dill, ¼ teaspoon salt, and ⅛ teaspoon pepper. Add the potatoes and toss.
Shape the beef into 16 patties and season with ½ teaspoon salt and ¼ teaspoon pepper.
Cook the patties in a large skillet over medium-high heat, 2 minutes per side for medium.
In a bowl, combine the cucumber and yogurt. Fill the pitas with the patties and cucumber sauce. Serve with the potato salad.