TZATZIKI: Put the strainer over a medium bowl, then put the grated cucumber into the strainer and sprinkle on ¼ teaspoon salt. Let it sit up to 30 minutes, pressing with the back of a spoon occasionally to squeeze out as much liquid as possible.
In the other medium bowl, stir together the strained cucumber, Greek yogurt, garlic, fresh dill, and lemon juice. Add more salt to taste, then stir in the olive oil and black pepper. Cover and keep chilled.
Shape the meat into 6 to 8 burger patties, folding 1 to 2 tablespoons of feta cheese in the middle of each (optional, but good). Grill the burgers to desired doneness.
Remove from the grill and cover with foil for a 2 to 3 minutes to seal in the juices. Assemble the burgers in the following order: bun bottom, burger patty, pesto, tzatziki, onion, bun top.