Prep the zoodles. Spiralize the zucchini, then lay the noodles on a clean kitchen towel and sprinkle lightly with salt. Let them sit for 10 minutes to draw out moisture while you cook the chicken. Blot dry before using.
Cook the chicken. Heat olive oil in a grill pan over medium-high heat. Add the chicken breasts and cook for 6-7 minutes per side, until internal temperature reaches 165°F. Transfer to a plate, cover with foil, and rest for 5 minutes before slicing.
Make the Alfredo sauce. In a saucepan, melt butter over medium heat. Add garlic and sauté for 1 minute until fragrant. Pour in the heavy cream, stir in Parmesan, and whisk until smooth. Simmer for 3-4 minutes until slightly thickened. Season with salt, pepper, and a pinch of red pepper flakes if using.
Combine zoodles and sauce. Add the dried zoodles to the saucepan. Toss gently for 1-2 minutes until heated through but still al dente. Overcooking will make them mushy!
Plate and serve. Divide the zoodle Alfredo between bowls, top with sliced chicken, and garnish with parsley, extra Parmesan, and a drizzle of olive oil.