For the buttery lemon noodles: Cook the pasta according to the package directions. Drain and set aside.
For the chicken piccata: Sprinkle the chicken breasts with salt and pepper on both sides, then dredge in the flour.
Heat the butter and oil in a large skillet over medium heat. Fry the chicken breasts until golden brown on both sides, about 3 minutes on each side; fry them a little longer if the breasts are thicker, monitoring the oil/butter mixture to make sure it doesn't burn. Remove the chicken to a plate.
Add the garlic to the skillet and saute. Pour in the chicken broth and wine, and squeeze in the juice of the lemon. Whisk the sauce, scraping the bottom of the skillet. Allow the sauce to cook and bubble and thicken until reduced by about half. Sprinkle in a little salt and pepper as it's cooking.
Reduce the heat to low and whisk in the cream. Add the capers, then cook for a couple of minutes until the sauce thickens. Taste and adjust the seasonings. Expect the sauce to have a real tang to it; counter it with a little more broth and cream if it's too strong!
To finish the noodles, melt the butter in a separate skillet over medium-high heat. Throw in the cooked pasta, along with the parsley. Cook it in the butter for a couple of minutes so that a few of the noodles get a little bit of a pan-fried texture to them. Grate in some lemon zest and squeeze in some lemon juice. Season with salt and pepper.
Place a handful of arugula on each plate, some noodles and a chicken breast. Garnish with Parmesan shavings and chopped parsley, then spoon over the sauce.