Add zoodles to a large bowl. Using kitchen shears snip zoodles into 3-4-in. lengths. Add spinach.
Cut chicken horizontally to a thickness of about 1/2-inch. Season with salt and pepper. In a very large skillet, heat 1 Tbs. olive oil over medium-high heat. Add chicken and cook for 2 minutes per side. Cover and reduce heat to low. Cook for 10 minutes (165°). Transfer chicken to a cutting board. Cool and cut into bite-sized pieces.
Add chicken and pesto to zoodle mixture. Toss to combine. Top with the shaved Parmesan cheese and basil leaves. Serve chilled or at room temperature.
The salad can be assembled without the pesto and refrigerated up to 24 hours before the potluck. Toss with pesto just before transport