Prepare a charcoal or gas grill for bi-level cooking, so that there is medium-high direct heat available on one side, and indirect on the other. (See note.)
Coat the cut sides of the bread with 2 tablespoons of the olive oil and season with 1/2 teaspoon of the salt. Grill directly over the coals, cut-side down for 1-2 minutes, or until nicely charred.
Meanwhile, place the kale leaves on an 18' x 24' piece of heavy-duty aluminum foil, brush with the remaining oil, and season with 1/4 teaspoon of the salt. Pile to one side of the foil.
When pleasantly charred, move the bread to the foil and grill the kale for 2-3 minutes, turning frequently, until slightly charred. Remove to a cutting board and when cool enough to handle, cut out the stems with a paring knife.
Spread the pesto evenly on the cut sides of the bread, then smear the cheese onto the bottom piece of the bread. Evenly shingle on the soppressata, kale, then the tomatoes. Season with the remaining salt and a few grinds of black pepper. Complete the sandwich with the top piece of bread and tightly wrap in foil.
Move the foil packet to the cooler side of the grill, close the lid and cook for 10 minutes. Carefully remove to a cutting board and rest for at least 5 minutes before unwrapping, slicing, and serving.