Slice garlic and chop the bottom ends of the asparagus spears off (usually 1-2 inches). Then chop spears into about 2 inch pieces.
Heat a large skillet over medium heat. Once hot, add a drizzle of oil along with the thin asparagus spears. Cook until they are bright green and are slightly tender, 4-5 minutes. Add garlic slivers and season with salt and pepper. Cook for another minute. Add white wine to deglaze pan and once it is cooked off, remove asparagus from skillet.
In a medium bowl, whisk together eggs with Parmesan cheese.
Cook pasta according to instructions until al dente. Reserve two cups of pasta water. Whisk 1/2 cup of hot pasta water into the egg/cheese mixture to lighten it a bit.
Drain pasta and add pasta to skillet with 1/4 cup pasta water over low heat. Fold in asparagus mixture.
Pour egg mixture over the pasta and asparagus and stir rapidly to mix in the eggs off the heat. The goal is to make it into a smooth sauce on the pasta. If the pasta gets too thick, add more pasta water by the 1/4 cup. If you do this on the heat, the sauce will overcook.
After a few minutes the egg mixture should barely hold onto the pasta as a sauce. Add fresh lemon juice to the pasta and season with salt and pepper.
Serve carbonara immediately with lemon zest and extra grated Parm on top.