Place the spiralized zucchini in a colander, toss with 1/2 tsp salt, and let drain for 15 minutes. Press out excess water with paper towels.
Season the sliced chicken with the remaining salt, pepper, and Italian seasoning.
Heat the olive oil in a large skillet over medium-high heat. Cook the chicken for 5-6 minutes until golden and cooked through, then transfer to a plate.
Reduce heat to medium and melt the butter in the same skillet. Add the garlic and sauté for 30 seconds.
Pour in the heavy cream and bring to a gentle simmer. Whisk in the parmesan and nutmeg until smooth and thickened, about 3 minutes.
Add the zucchini noodles and toss for 1-2 minutes until just tender. Return the chicken to the pan and stir to coat.