Combine the bread pieces and milk in a medium bowl. Mash with a fork until a paste forms. Add the shallots, garlic, mint, oregano, salt, and pepper; mix well. Add the lamb, then use your hands to mix until well combined. Form the meat mixture into 6 oval-shaped patties about ½-inch thick.
Oil the grilling grates. Grill the patties, covered, until nicely browned on the first side, 2 to 4 minutes. Flip the burgers and cook for a few minutes more until desired doneness is reached. Place burgers on a tray and cover with foil while you warm pita rounds on the grill. Assemble the burgers and pass toppings and tzatziki sauce alongside.
Note: There are two types of pita bread: pita pockets and pocketless pitas. You can use either for this recipe. For pita pockets, be sure they are at least 6 inches wide. Trim off the top ¼ of each round and stuff the burgers and toppings inside. For pocketless pitas, serve burgers open-faced or, if pitas are large enough, wrap them around burgers.
Make-Ahead/Freezer-Friendly Instructions: The uncooked burgers can be made and refrigerated up to one day ahead of time or frozen for up to three months. (Freeze the burgers on a baking sheet or plate so their shape sets, then transfer them to a sealable plastic bag for easy storage.) Defrost the burgers overnight in the refrigerator prior to serving and then cook as directed.