In a 2-cup liquid measuring cup, whisk together the warm water, sugar and yeast. Let the mixture stand 8 to 10 minutes, until it turns pale and foamy on top: that foam is the proof the yeast is alive.

In a large mixing bowl, whisk the flour and kosher salt together. Pour in the yeast mixture and 1/4 cup of the olive oil, then stir with a wooden spoon until a shaggy, sticky dough comes together with no dry flour left in the bottom of the bowl.

Cover the bowl with a damp towel and set it somewhere warm. Let the dough rise for 1 to 1 1/2 hours, until it has doubled in size and holds a dent when you poke it with a floured finger.

Pour 2 tablespoons of the remaining olive oil into a white ceramic baking dish, about 9 by 13 inches, and tilt it to coat the bottom and sides. Scrape the dough into the dish and press it toward the corners with oiled fingers; if it springs back, rest it 10 minutes and press again until it fills the dish evenly.

Cover the dish loosely and let the dough rise a second time for 30 to 45 minutes, until it looks puffy and has risen almost to the rim.

Press your fingertips straight down all over the surface to make deep dimples. Drizzle the remaining 2 tablespoons olive oil over the top, letting it pool in the dimples, then scatter with the rosemary and flaky salt.

Bake in a 425°F oven for 22 to 26 minutes, until the top is deep golden brown and the edges have pulled slightly away from the dish. It is done when tapping the center with a knuckle sounds hollow.

Run a thin knife around the edge of the dish and lift the focaccia onto a wire rack. Let it cool completely, at least 45 minutes: warm bread tears instead of slicing cleanly.

Set the cooled focaccia on a large wood board. Using a long serrated knife held flat, slice it in half horizontally all the way through to make two even slabs.

In a small bowl, whisk the mayonnaise and pesto together until smooth and pale green. Spread the mixture evenly over the cut side of both focaccia halves, right to the edges.

On the bottom half, layer the provolone, then the deli meats, then the roasted peppers, then the arugula. Season the arugula lightly with salt and a few turns of black pepper.

Set the top half in place, pesto side down, and press firmly and evenly with both hands across the whole surface. Wrap the sandwich tightly in plastic wrap, then set another dish or a cutting board on top weighted with a couple of heavy cans.

Refrigerate the pressed sandwich for at least 1 hour, or up to overnight, so the layers settle together. It is ready to cut as soon as it feels firm and slightly compressed through the wrap.

Unwrap the sandwich and trim the ragged edges with a serrated knife. Cut into 24 even squares, a grid of six by four, and arrange on a large white ceramic platter to serve.