Halve and peel shallot. Slice half the shallot into thin strips and finely mince other half. Place shallot strips in a mixing bowl and cover with cold water. Set aside 20 minutes. Soaking shallots tames their sharp flavor. While shallot sits, cut potato into eight wedges. Grate one cucumber on large holes of grater. Trim other cucumber and cut into 1/4" rounds on an angle. Mince dill, leaves and stems.

Place potato wedges on prepared baking sheet and add 2 tsp. olive oil, 1/4 tsp. salt, and a pinch of pepper. Use your hands to coat potato and spread into a single layer. Roast until tender and easily pierced, 18-22 minutes, flipping wedges once after 10 minutes. While potatoes cook, make gyro mix.

Thoroughly combine beef, panko, half the minced shallot (reserve remaining for tzatziki), 1 Tbsp. yogurt (reserve remaining for tzatziki), cumin, 1/2 tsp. salt, and a pinch of pepper in another mixing bowl.

Form gyro mix into six golf ball-sized meatballs. Place a medium oven-safe pan over medium heat and add 2 tsp. olive oil. Add meatballs to hot pan and cook until browned, 6-8 minutes. Place pan in oven and bake until meatballs reach a minimum internal temperature of 145 degrees, 8-10 minutes. While meatballs bake, mix remaining yogurt, grated cucumber, remaining minced shallot, dill, and a pinch of pepper in another mixing bowl.

Place flatbreads directly on oven rack and toast until warmed, 3-5 minutes.

Spread a dollop of tzatziki sauce on flatbread. Serve remaining sauce on side for dipping. Place meatballs, cucumber rounds, and shallot slices in flatbread. Garnish fries with feta. Place sandwich and fries on a plate.