Preheat oven to 425 degrees.
Place whole chicken, breast side up, in a large cast iron skillet, or other large baking pan.
Rub 2 tablespoons of the olive oil over the bird. Season outside and inside with kosher salt and freshly ground black pepper. Place fresh parsley stems and a few of the lemon quarters inside the chicken cavity. Tie legs together with kitchen twine. Tuck wings up under the chicken.
Toss the potatoes and carrots in 2 tablespoon of the olive oil. Season with kosher salt and freshly ground black pepper. Scatter around the chicken. Squeeze the remaining lemon quarters over the chicken and the vegs. Toss the squeezed lemons into the pan as well to roast.
Roast for 15 minutes. Reduce heat to 375 degrees and cook for 25 minutes. Rotate the pan, toss the vegs and cook until the vegs and chicken are golden brown and a thermometer inserted into the thickest part of the chicken reads 165 degrees, about 25 minutes more. Remove from oven and let chicken rest for 10 minutes.
While chicken is roasting make the sauce using a mini chopper or by hand. Combine minced garlic cloves, chopped parsley, 1/2 cup remaining olive oil, grated parmesan cheese, salt to taste and a pinch of red pepper flakes to taste. Blend until mixture is like a rough-chop pesto. Taste and adjust seasonings if need be.
Drizzle the sauce over the roasted chicken and vegs. Toss potatoes and carrots to cover well. You can reserve some of the sauce to serve on the side when the chicken is carved. Amazingly fresh, juicy and Spring-like roasted chicken.