For the aioli: Place the yolk in a stainless steel bowl. Begin whisking in the grapeseed oil drop by drop. Once the mixture has thickened and emulsified, you can whisk in the remaining grapeseed and olive oils in a slow steady stream. If the mixture gets too thick, add a drop or two of water.
Pound the garlic with 1/4 teaspoon salt with a mortar and pestle. Whisk the garlic paste into the aioli. Season with 1/4 teaspoon salt, a squeeze of lemon juice, and the cayenne. Taste for balance and seasoning. If the aioli seems thick and gloppy, thin it with a little water. In addition to thinning the aioli, this will also make it creamier.
For romesco: Preheat the oven to 375° Remove and discard the stems and seeds from the chiles, and then soak them in warm water for 15 minutes to soften. Strain the chiles, and pat dry with paper towels.
Meanwhile, spread the nuts on a baking sheet and toast for 8 to 10 minutes, until they smell nutty and are golden brown.
Heat a large sauté pan over high heat for 2 minutes. Add 2 tablespoons olive oil and wait a minute. Fry the slice of bread on both sides until golden brown. Remove the bread from the pan and cool. Cut it into 1-inch cubes and set aside.
Return the pan to the stove over high heat. Add 2 tablespoons olive oil and the chiles and sauté for a minute or two. Add the tomatoes. Season with 1/2 teaspoon salt and cook 2 to 3 minutes, stirring often, until the tomato juices have evaporated and the tomato starts to color slightly. Turn off the heat, and leave the mixture in the pan.
In a food processor, pulse together the toasted nuts, garlic, and fried bread until the bread and nuts are coarsely ground. Add the chile-tomato mixture and process for a minute more.
With the machine running, slowly pour in the remaining 1 cup olive oil and process until you have a smooth purée. Don’t worry, the romesco will “break” or separate into solids and oil; this is normal. Add the parsley, and season to taste with lemon juice and more salt if you like.
For the Burger: In a medium sauté pan, toast the cumin seeds over medium heat a few minutes until the seeds release their aroma and darken slightly. Pound the seeds in a mortar or spice grinder until coarsely ground.
Return the pan to the stove over high heat for 1 minutes. Add the olive oil and shallots. Turn the heat down to medium-low, and cook for a few minutes, sitrring, once or twice, until the shallots start to soften. Add the garlic, thyme, cumin and sliced chile. Season with 1/4 teaspoon salt and a few grindings of black peppery, and cook 3 to 4 minutes, until the shallots become translucent. Set aside to cool.
In a large bowl, use your hands to combine the ground pork, chorizo, bacon, shallot mixture, and parsley, being careful not to overmix the meat. Season with 1 1/4 teaspoons salt and lots of freshly ground black pepper. Shape the meat into six 6-ounce patties. Chill in the refrigerator if not using right away.
Light the grill 30 to 40 minutes before cooking and remove pork burgers from the refrigerator to come to room temperature (if you made them in advance).
When the coals are broken down, red, and glowing, brush the pork burgers with olive oil and grill them 3 to 4 minutes on the first side, until they’re nicely browned. Turn the burgers over, and place a piece of cheese on each one. Cook another 3 minutes or so, until the pork is cooked through. (It should still be slightly pink in the center.)
Slice the buns in half, brush them with olive oil, and toast them on the grill, cut side down, for a minute or so, until they’re lightly browned.
Spread both sides of the buns and the aioli. Place a burger on the bottom half of each bun, and dollop with a generous amount of romesco. Place some arugula leaves on top, and finish with the top half of the bun.