Step 1Roast the chiles and peppers: Heat the oven to 400 degrees. Place the sliced chiles on a rimmed baking sheet and drizzle over 2 tablespoons olive oil. Sprinkle over a pinch of kosher salt and toss to coat the chiles with oil and seasoning. Roast the chiles until slightly wilted, 4 to 6 minutes. Remove and cool the chiles. Decrease the oven temperature to 350 degrees to roast the peppers.
Step 2To roast the peppers, place them on a rimmed baking sheet. Rub them all over with 2 tablespoons olive oil and season with one-fourth teaspoon kosher salt. Roast the peppers, turning them occasionally, until their skins are wrinkled and slightly charred in places, about 30 minutes. Remove and cool the peppers to room temperature. Increase the oven temperature to 450 degrees.
Step 3Remove the dishcloth from the top of one of the focaccias and, using your fingertips, gently tap down on the focaccia with about 5 light strokes to nudge it toward the edge of the pan; it still might not reach the edges, but don't worry.
Step 4One at a time, push the mozzarella cubes into the focaccia dough while simultaneously pushing outward to encourage the dough toward the edge of the pan, arranging the cubes evenly over the surface of the dough and pressing them so deep they are almost flush with its surface. Do the same with the tomatoes.
Step 5Sprinkle over the thinly sliced chiles, using your palms to gently pat them flush with the top of the dough, over the other toppings.
Step 6Arrange the peppers: Place 1 to 3 peppers on the surface of the dough, arranging them in a rough circle approximately 3 inches in diameter, overlapping slightly if needed. Using the tips of your fingers, gently press the peppers into the dough, rotating your hand clockwise (or counterclockwise) to form the peppers into a loose flower or rosette shape. Be sure not to press the peppers too far to the bottom of the pan. Repeat, alternating the pepper colors, to form approximately 8 rounds over the surface of the dough.
Step 7Sprinkle over the thyme and sea salt. Set the focaccia aside until it has risen and puffed around the toppings, about 30 minutes.
Step 8Place the focaccia on the center rack of the 450-degree oven and bake until crisp and golden-brown, 30 to 40 minutes. The focaccia should have risen almost to the top of the pan, the peppers will be charred and the cheese will be crisp and browned.
Step 9Move the pan to the bottom of the oven and continue to bake until the bottom crust of the focaccia is deep golden and crisp to the touch, about 5 minutes more.
Step 10Remove the pan from the oven and remove the focaccia from the pan to a wire rack (use a fork to gently lift and slide the focaccia out of the pan, careful of any hot oil that may still be at the bottom of the pan). Brush the surface of the focaccia once more with olive oil. Set the focaccia aside to cool slightly -- or as long as you can resist it.
Step 11Transfer the focaccia to a cutting board. Halve the focaccia using a long knife, then cut one-half into 4 wedges (don't cut the other half until you are ready to eat it).