Combine marinating ingredients and chicken in a gallon zippered bag. Place in refrigerator and let marinate for around 4 hours (up to 24hrs).
Preheat your grill (or grill pan) to medium-high. Start water boiling for pasta.
In a large saute pan over medium heat, add butter. Once melted add garlic, carrot ribbons, salt and pepper. Stir to coat and let saute until carrots are tender. Reduce heat to medium-low and add heavy cream and dill. Allow to thicken, stirring occasionally.
Once grill is hot, start cooking chicken breasts (discard marinating liquid). Cook for 8-10 minutes on each side (until thickest part of thickest piece reaches internal temp of 165degF). Remove to cutting board and let rest.
Add spaghetti to boiling water and cook as directed on the package.
Add zucchini ribbons and parmesan cheese to carrots and cream sauce. Stir to combine. Let cook another 3 minutes.
Drain pasta in colander and then add to cream sauce. Toss to evenly coat.
Slice chicken, as desired.
Serve portion of pasta with sliced chicken on top.