For the flatbread: Combine the yeast, sugar and 1/4 cup lukewarm water in a small bowl and let the yeast puff up and proof.
Pour the yeast mixture into a large bowl and add 1 cup lukewarm water, the oil and salt. Add 1 cup of the flour and whisk until smoothly combined. Switching to a wooden spoon, add the rest of the flour and stir until you have a soft dough. Rub the dough with a bit more oil, press a square of plastic wrap to its surface and let it rise at room temperature until doubled in size, about 1 hour.
Punch down the dough, transfer it to an oiled plastic bag and let it proof in the refrigerator at least 4 hours or as long as 3 days.
For the peppered pork loin: To give the roast a quick cure for added flavor, stir together the sugar and 1/2 teaspoon salt, rub the mixture all over the roast and refrigerate for 1 hour or as long as overnight.
Preheat the oven to 375 degrees F.
Blot the roast dry. Crush the peppercorns with a mortar and pestle (or in a spice-devoted coffee grinder) to a medium-coarse texture. (Some will be finely ground, but you want the larger pieces to resemble cracked pepper.) Rub the pepper into the meat, saving what doesn't stick for the marinade.
Heat a small saucepan over medium-low heat, and add the butter and garlic. Cook until the butter bubbles; add the vinegar, thyme and remaining black pepper, and remove from the heat.
Sprinkle the pork with 1 teaspoon salt. Heat an ovenproof skillet large enough to fit the pork roast over high heat and add the oil. Brown the meat quickly on all sides.
Brush the meat generously with some of the pepper mixture, put the skillet in the oven and roast for 10 minutes. Remove the skillet, roll the roast to another side and baste it with more of the pepper mixture. Continue to roast the meat, removing it from the oven every 10 minutes or so to flip and mop, until an instant-read thermometer inserted in the center of the roast reads 145 degrees F, about 50 minutes total.
Transfer the meat to a platter and let it rest for at least 10 minutes, rolling the pork around to sop up its exuded juices. Refrigerate the pork until using. (The pork could also be sliced and served hot at this point.)
For the doctored aioli: Stir together the mayonnaise, olive oil, salt, pepper, garlic and lemon zest and juice in a medium bowl. Refrigerate until using.
Heat a grill over high heat. Generously oil a baking sheet. Divide the flatbread into eight portions on the baking sheet. Press the dough into thin rounds, about 8 inches in diameter.
Grill the breads over medium-high heat, closing the lid right after putting them on and flipping and turning to avoid any flare-ups. Each flatbread will take about 2 minutes to cook.
Slice the pork very thinly. Spread some aioli across the flatbreads and pile some pork on one side of each. Top with shaved Parmesan and a handful of arugula. Fold the sandwiches in half and pile them on a platter for serving.