Step 1: Combine the beef, veal, garlic, onion, serrano peppers, parsley, salt, and black pepper in a large mixing bowl. Knead the mixture gently with your hands until thoroughly blended. Cover the bowl with plastic wrap and refrigerate for 30 minutes to 1 hour.
Step 2: Divide the meat into four equal portions. Mold each into a large flat disk, like a pita bread. Each pljeskavica should be 6 to 8 inches across and 1/4-inch thick. In Bosnia, they’d mold and flatten the burgers right on the work counter. (You pry them off with a long spatula.) You may find it easier to flatten the burgers between two sheets of parchment paper, waxed paper, or plastic wrap. Chill the burgers until ready to grill.
Step 3: Set up your grill for direct grilling and preheat to high. Brush and thoroughly oil the grill grate.
Step 4: Loosen the burger from the baking sheet by sliding a slender metal spatula or carving knife under it, or peel it off the waxed paper. Carefully transfer it to the grill. Grill until cooked through, 3 to 4 minutes per side, turning with a wide-headed spatula. In this part of the world, burgers are served medium to medium-well.
Step 5: Transfer the pljeskavicas to a platter or plates and serve at once with pita, tomato slices, pickled peppers, and lemon wedges for squeezing.