Prepare a grill by adjusting the grill grate so that it's 4 to 5 inches from the flame or heat source. Heat a gas grill to high or a charcoal grill to medium hot (the coals should be covered with light ash, and you should be able to hold your hand just over the grate for no more than three seconds).
Grill the tomatoes, bell peppers, and jalapeño, turning them occasionally, until the tomato skins are cracked and blackened in spots, 3 to 4 minutes, over charcoal, about twice that for gas, and until the pepper and jalapeño skins are blackened, 7 to 8 minutes. Remove them from the grill. Let the tomatoes cool and put the peppers and jalapeño in a paper or plastic bag to steam until cool enough to handle, about 10 minutes. Remove and discard the skins, seeds, and membranes from the peppers and jalapeño but not the tomatoes.
In a blender or a food processor, combine the tomatoes, peppers, jalapeño (as well as the juices from the vegetables), garlic, vinegar, and 3 Tbs. of the olive oil and pulse until the salsa has a chunky consistency. Season to taste with salt and pepper.
In a medium bowl, whisk the lemon juice with 1 Tbs. of the olive oil. Season with salt and pepper.
Brush the chicken cutlets with 1 to 2 Tbs. olive oil. Grill until golden on one side, about 2 minutes, turn them over, season with salt and pepper, and continue to grill until golden and cooked through, another 2 minutes.
Add the arugula and shaved Parmigiano to the lemon juice and olive oil and toss.
To serve, divide the chicken cutlets among six plates and top with the salsa and then with the arugula salad. Serve immediately.