Bring a large pot of water to a boil and add 2 tablespoons of salt. Stir to dissolve and add the pasta. Cook the pasta according to package directions and drain, reserving 1 cup of pasta water before draining. Do not rinse the cooked pasta. Set aside.
Warm the olive oil in a Dutch oven or deep-sided skillet over medium-high heat. Add the chicken, toss to coat, and sprinkle generously with salt and pepper. Cook for 4-5 minutes until cooked through. (If using breasts, watch closely and reduce cooking time as needed.) When the chicken is done cooking, transfer to a bowl and set aside.
In the same pot or pan used for the chicken, melt the butter and add the garlic. When the garlic is fragrant, add the cream and bring to a simmer over medium to medium-low heat. Add the lemon juice and lemon zest and whisk to combine.
Cook for 2-3 minutes, until slightly thickened. Add the parmesan and stir to melt. Add the pasta, and the peas if desired. Toss to coat.
Add a splash or two of the reserved pasta water, if the pasta appears dry. Taste and adjust the salt and pepper, if desired. Sprinkle with parsley and stir once more. Top individual servings with additional Parmesan. if desired.