Heat grill over medium-high heat (350-400℉) and oil the grill grates. To do this, dip a wadded paper towel in a little avocado oil and, using tongs, wipe the oil evenly over the grate. Be careful not to use too much oil, because that's a sure fire way to start a good flare-up—a little goes a long way here.
Add the chicken to one half side of the grill and cook for 4 minutes per side, until the pieces turn golden and grill marks appear. The lid should be closed throughout the cooking process.
To the other half of the grill, add the mini bell peppers and zucchini, in a single layer. Grill for about 6 minutes total, flipping halfway through, until grill marks appear and the zucchini is tender and the bell peppers soften. Transfer to a serving platter and set aside.
Reduce the heat to medium-low and divide the marinara on top of each of the chicken breasts. Next, top with the mozzarella, then the giardiniera. Close the lid and let cook until the cheese is nice and melty, 3 to 4 more minutes.