Spiralize zucchini into noodles using the blade that creates spaghetti-width strands. Spread on a clean kitchen towel, sprinkle with 1 teaspoon fine sea salt, and let sweat for 15 minutes. This draws out excess water so your sauce won't pool. Pat very dry with additional towels before cooking.

Heat olive oil in a large skillet over medium-high heat until shimmering. Season chicken with 1/2 teaspoon kosher salt and pepper. Add to skillet and cook 6-7 minutes per side until golden brown and internal temperature reaches 165 degrees F. Transfer to a plate, tent with foil, and rest 5 minutes. Slice against the grain into 1/2-inch strips.

Reduce heat to medium. Add butter to the same skillet and melt, scraping up any browned bits. Add garlic and cook 30 seconds until fragrant but not browned - it turns bitter fast.

Pour in heavy cream and bring to a gentle simmer. Cook 4-5 minutes, stirring occasionally, until slightly thickened and coats the back of a spoon. The sauce will thicken more as it cools, so stop when it's a touch looser than you want.

Remove from heat. Whisk in Parmesan, lemon juice, remaining 1/4 teaspoon kosher salt, and nutmeg if using. Stir until smooth and glossy. If sauce seems thick, thin with 1-2 tablespoons pasta water or additional cream.

Add dried zucchini noodles to the sauce and toss over medium-low heat just 2-3 minutes until warmed through and barely tender. They should still have bite - overcooked zoodles release water and turn mushy. Tongs work best here.

Divide zoodles and sauce among shallow bowls. Top with sliced chicken, extra Parmesan, and chopped parsley. Serve immediately while the sauce is still loose and creamy.