Heat the oil in a large skillet over medium-high heat. Sprinkle both sides of the chicken with the dried Italian seasoning and a good pinch of salt. Arrange the chicken in the pan, cover and sear for 5 minutes. Flip, cover and sear the chicken another 5 minutes. (It won't be cooked all the way through, but that's okay! We'll finish in a bit.) Remove the chicken from the pan.
Add a splash of water to the pan to deglaze, if needed, and use a wooden spoon to scrape up the browned bits. Add the zucchini to the pan in a single layer. Season with a pinch of salt and sear for 2 minutes. Flip and sear until nicely charred on both sides, another minute or two. Add the chicken (and its juices) back to the pan, and bring the heat down to medium-low. Add the tomato sauce, along with about 1/3 cup water swirled in the jar, along with the gnocchi. Using tongs, nestle the gnocchi down in the sauce, so that it's all covered. Cover with a lid and bring the heat up to medium. Simmer until the gnocchi is tender, 5 to 6 minutes.
Divide the dish among dinner bowls, and use forks to shred the chicken a little, if you want. Garnish with freshly grated Parmesan and fresh basil leaves.