Make the spiced butter: Melt the butter over low heat in a small saucepan. Add the harissa and garlic and cook 1 minute. Remove from the heat, cover and let sit 1 hour to allow the flavors to meld.
Make the pickled saffron shallots: Bring the vinegar, sugar, 1 teaspoon salt and the saffron to a boil in a small saucepan set over medium heat; remove from the heat and let cool 10 minutes. Put the shallots in a small bowl, pour the vinegar mixture over them, cover and refrigerate at least 30 minutes and up to 24 hours.
Make the aioli: Combine the mayonnaise, harissa, smoked paprika, piquillo peppers, anchovy and lemon juice in a blender and blend until smooth; season with salt and pepper. Cover and refrigerate at least 30 minutes to allow the flavors to meld.
Make the crispy prosciutto: Heat the canola oil in a small saute pan over medium heat until it shimmers. Add 1 slice of the prosciutto at a time to the pan and cook until just crisp. Drain on a plate lined with paper towels.
Make the burgers: Combine the chuck and fat in a meat grinder and grind coarsely into a large bowl (or put the ground beef in a large bowl). Mix in some salt and pepper. Divide the meat into 4 equal patties.
Heat a cast-iron pan over high heat until smoking. Add the canola oil and burgers and cook until the bottoms are golden brown and a crust has formed, about 4 minutes. Flip over and begin brushing with the spiced butter. Cook until the bottoms are golden brown and the meat is cooked to medium, 4 more minutes. Top the burgers with the fontina, add a little water to the pan, cover quickly and cook until the cheese is completely melted, about 1 minute.
Spread some aioli on the top and bottom buns. Place a burger on each bottom bun; top with some of the pickled shallots, followed by a slice of crispy prosciutto and some arugula. Add the top buns and serve.