In a plastic or ceramic bowl or crock, stir the 1 quart water, 1/4 cup kosher salt, 1/4 cup sugar, and 1 cup vinegar together until the salt and sugar have dissolved (this will take a few minutes). Cut the chicken into large chunks. Immerse in the brine and refrigerate overnight.

Remove the meat from the brine and rinse under cool running water. Pat dry with a towel. In a deep baking dish or roasting pan, combine the marinade ingredients - the olive oil, garlic, thyme, bay leaves, rosemary, shallots, and a little salt and pepper. Thread the pieces of meat onto 4 wooden skewers and immerse the skewers in the marinade. Cover with plastic wrap, pressing the plastic right down onto the meat to exclude any air. Refrigerate for at least 4 hours; overnight is even better, though not essential.

Remove the skewers from the refrigerator and let stand at room temperature for 20 to 30 minutes while you preheat a charcoal or gas grill, or ridged cast-iron grill pan, until hot.

Prepare the Greek salad: rush the onion slices with a little olive oil and season with kosher salt and pepper. On a griddle pan or cast-iron skillet, grill the onions until tender. Separate into rings.

In a large bowl, combine the remaining salad ingredients.

Make the Ladolemono: In a bowl, combine the lemon juice, mustard, oregano, 1/2 teaspoon kosher salt, and a generous grinding of pepper. Whisk to blend the mixture completely and, whisking all the time, drizzle in the olive oil. This sauce will separate; whisk or shake in a jar before using.

Make the Tzatziki: Quarter the cucumber lengthwise and trim off the triangular wedge of seeds. Cut the cucumber into very small, even dice. Transfer to a mixing bowl. In a food processor, combine the garlic, vinegar, shallots, and dill. Pulse until finely chopped but not pureed. Add the mixture to the cucumbers; add the yogurt. Fold together with a rubber spatula, adding the olive oil and lemon juice. Season liberally with kosher salt and pepper, starting off with 1 tablespoon salt. Taste for seasoning. (You can store Tzatziki in a covered, clean jar in the refrigerator for up to 1 week.)

Season the Souvlaki lightly with salt and pepper and grill until firm and char-marked on all sides, about 6 minutes. Transfer to a resting platter and dress with Ladolemono.

Paint the pita with extra-virgin olive oil, season with salt and pepper, and grill until firm and char-marked on all sides. Smear some Tzatziki in the center of each pita round. Push the meat off the skewer on top of the tzatziki. Top with some of the Greek Salad. Pull the sides of the pita up to meet in the center, like a taco. Serve like a hand roll, or wrap and secure with a wide strip of parchment.