Heat oven to 425° F. Toss the potatoes, 2 tablespoons of the oil, and ¼ teaspoon each salt and pepper on 2 rimmed baking sheets. Roast, tossing once, until golden brown and cooked through, 22 to 25 minutes.
Meanwhile, gently mix together the turkey, scallions, cumin, and ¼ teaspoon each salt and pepper in a medium bowl with your hands until just combined. Form the mixture into four ½-inch-thick patties.
Heat the remaining 1 tablespoon of oil in a large nonstick skillet over medium-high heat. Cook the patties until an instant-read thermometer inserted in the center of a patty registers 165° F, 5 to 7 minutes per side. Wipe out the skillet; reserve.
Brush both sides of the bread with the mayonnaise. Sandwich the patties and cheese between the bread. Cook, in 2 batches, covered, in the reserved skillet over medium-low heat until the bread is golden brown and the cheese is melted, 1 to 3 minutes per side.
Serve the patty melts with the oven fries, carrot sticks, and cucumber spears and the salad dressing for dipping.