2 large eggs, lightly beaten
1 1/2 teaspoons kosher or coarse sea salt, or to taste
1/2 teaspoon freshly ground black pepper, or to taste
Mexican Manchego or Monterey Jack cheese slices (optional)
Hamburger buns, for serving
For the aioli:
1 cup regular or low-fat mayonnaise
1 teaspoon finely grated zest and 2 tablespoons fresh lime juice
3 cloves garlic, minced or pressed
1 teaspoon kosher or coarse sea salt, or to taste
1/4 teaspoon freshly ground black pepper, or to taste
For the burgers: Prepare the grill for direct heat. If using a gas grill, preheat to medium-high (450 degrees). If using a charcoal grill, light the charcoal; when the coals are ready, distribute them evenly over the cooking area. For a medium-hot fire, you should be able to hold your hand about 6 inches above the coals for 4 or 5 seconds. Use cooking oil spray to grease the grill grate and place on the grill. Have ready a spray water bottle for taming any flames.
Place the dried chilies in a small bowl and cover with boiling water. Soak for 10 to 15 minutes. Use a slotted spoon to transfer them to a blender, then add 2 tablespoons of the soaking liquid (discarding the rest), the onion and garlic. Puree until smooth.
Combine the ground beef and veal in a large mixing bowl. Mix in the chili puree, eggs, salt and pepper. Use your hands to shape the meat into 8 to 10 large patties.
Arrange the patties on the grill; cook uncovered for 5 minutes per side (medium-well).
Meanwhile, make the aioli: Whisk together the mayonnaise, lime zest and juice and garlic in a medium bowl. Season with salt and pepper to taste. The yield is about 1 cup. Cover and refrigerate until ready to serve.