Build a fire and let burn to ashy chunks (see "The DIY Firepit," below).
Preheat oven to 400°. Cut tops off heads of garlic, sprinkle with salt and pepper, and drizzle with olive oil. Wrap in foil and roast in oven for 1 hour. Let cool, then squeeze cloves from skin. (For Chiarello's Ash-roasted Garlic recipe, go to sunset.com/firepitmenu.)
Meanwhile, make pesto: Boil a large pot of water on the stove and salt generously with kosher salt. Boil asparagus about 3 minutes, or until tender; drain and spread out to cool.
Cut asparagus into thirds and save tips for salad. In a food processor, pulse together asparagus stalks, pine nuts, basil, garlic, sea salt, and pepper to taste. With machine running, drizzle in 6 tbsp. oil. Add parmesan in batches, pulsing after each batch (pesto should be thick). If cheese begins to clump, add water, 1 tsp. at a time, until it loosens. Cover with plastic wrap, smoothing it against surface of pesto.
Finish salad: In a small bowl, whisk 2 1/2 tbsp. oil, vinegar, 1/4 tsp. sea salt, and 1/8 tsp. pepper; set aside. Put asparagus tips, arugula, spring onions, roasted garlic cloves, and lemons in a large bowl.
On a floured surface, dust balls of dough with flour. Working with 1 ball at a time and keeping others covered, roll or stretch until 9 to 10 inches across. "If you have trouble rolling them out, let them rest 30 seconds to relax," says Chiarello. Draping the middle of the dough over the backs of both hands, lay it gently on grate. Brush with oil and sprinkle with salt and pepper.
Grill dough until it bubbles and is browned underneath, about 2 minutes. Using long tongs, flip over, brush with more oil, sprinkle with salt and pepper, and grill another 2 minutes, moving over lower heat if necessary (push coals to one side to make a lower-heat spot). Stretch and grill remaining dough rounds the same way. Don't worry if they're not perfectly round. "I like my crusts sort of free-form, deformato."
Top each piadina with about 1/3 cup pesto. Toss salad with dressing and divide among piadene. Top with pecorino and cut into slices, or fold and eat taco-style.
Make ahead: Pesto, 2 days chilled or 2 weeks frozen.
The DIY Firepit
"You can do anything on this. It's a little like camping in the middle of your day," says Napa Valley chef Michael Chiarello. All you need is bricks arranged to fit under your cooking grate and some sand.
Build the pit: Spread a double layer of heavy-duty foil on the ground ("not grass," warns Chiarello, since it may scorch). Make it big enough to extend a foot beyond your cooking grate in all directions. Build a brick rectangle 3 layers high, leaving a couple of bricks out of the top layer on opposite sides, to encourage airflow. (For a standard 21-in. round Weber grate, the rectangle should be 2 bricks by 3 bricks.) Fill with about an inch of sand.
Light the fire: Put several balled-up sheets of newspaper in the center and position kindling into a tipi around it; lean larger kindling and then 5 to 6 small logs (preferably oak). Light the fire. "The tipi lets every bit of flame go up past 3 or 4 logs," says Chiarello, so the fire starts fast. Once the logs have caught, add several larger logs to the perimeter. Let them burn down to ashy chunks with low flames (1 1/2 to 2 hours). Because there's less smoke and char than cooking over a flaming fire, Chiarello says, "it makes your food taste much cleaner, gives nuanced flavor, and is better for wine."
Start cooking: Keep another log burning at the back of the pit. When it's ashy chunks, rake it into the main fire to maintain heat.