Prepare the aioli: Combine mayonnaise, garlic, piquillo peppers, paprika, and salt in a food processor. Process until smooth.

Refrigerate the aioli for at least 30 minutes. It can be prepared 1 day in advance, tightly covered.

Divide the ground chuck into 4 equal portions (about 6 ounces each).

Form each portion into a 3/4-inch-thick burger and make a deep impression in the center with your thumb.

Season both sides of each burger with salt and pepper.

Heat a gas grill to high or heat coals in a charcoal grill until they glow bright orange and ash over.

Alternatively, heat a grill pan over high heat on top of the stove, or heat canola oil in a pan or griddle until it shimmers.

Brush the burgers with oil.

Grill the burgers until golden brown and slightly charred on the first side, about 3 minutes.

Flip the burgers and cook until golden brown and slightly charred on the second side, 4 minutes for medium-rare (3 minutes if topping with cheese) or until cooked to desired doneness.

Place the buns on a flat surface and spread half of the aioli over them.

Top each bun with a slice of Manchego cheese, then a slice of Serrano ham, then a burger, another slice of ham, and finally another slice of cheese, in that order.

Spread the rest of the aioli on the bun tops and cover each burger with a top.

Brush the tops of the buns with the remaining canola oil.

Place the burgers on the grill, griddle, or saute pan, top side down.

Using a heavy-duty metal spatula, press down on the bottom of the buns and grill until the tops are light golden brown, 1 to 2 minutes.

Turn the burgers over and continue cooking, pressing down the tops, until the bottoms are light golden brown and the cheese has melted, about 1 minute longer.

Serve immediately.