For the turkey burgers: Preheat a grill or grill pan over medium-high heat until hot. Clean and oil the grates.
Place the turkey, salt, pepper, feta and scallions in a medium bowl. Mix with your hands until combined. Form into 5 football-shaped patties.
Grill the patties until the internal temperature reaches 165 degrees F, 5 to 6 minutes per side. Remove the patties to a plate.
While the turkey patties cook, make the tzatziki sauce: Stir together all of the ingredients in a small bowl. Taste and adjust for seasoning, if necessary. Refrigerate until ready to serve, up to 3 days.
Lightly grill the pitas, 1 to 2 minutes per side.
To assemble: Cut the grilled pita pockets in half. On each of the bottom halves, place 4 to 5 avocado slices, then a turkey patty. Dollop with a generous amount of the tzatziki. Top with a couple of the lettuce leaves. Close the burgers and serve warm.
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