Preheat the oven to 425 F.
While the oven is preheating, place the chicken on a paper towel lined plate or a cutting board.
Dry the chicken with paper towels. This step is very important - if the chicken is not dry, it won’t brown. Season generously with salt and freshly ground black pepper on both sides. Melt the butter and olive oil in an ovenproof pan over medium heat for a few minutes.
Once the butter and oil are hot (it should take about 5 minutes, depending on your stove), carefully add the chicken breasts. Do not crowd the pan!
Sear for 4 minutes and flip over. It may feel stuck, but you can “wiggle” it gently using tongues or two spatulas, but don’t move it from side to side while it cooks. Sear the other 4 minutes as well.
Put the pan in the oven and bake for 20 minutes. Internal temperature should be 165 F minimum.
While chicken is in the oven, you have time to make the salad.
In a medium bowl, toss arugula with olive oil, freshly squeezed lemon juice, salt, and freshly ground black pepper.
Divide greens between two plates. Top with shavings of Parmigiano Reggiano, and half of an avocado per plate. *If you are not serving the salad immideately, squeeze additional lemon juice over avocado to prevent darkening. Once the chicken is cooked, let sit for at least 5 minutes. It’s a good rule for any meat - juices “run wild” in meats while it’s cooked, so letting it sit for a few minutes allows the juices to soak back in which is responsible for most of the flavor.
Add the chicken to prepared salad plates and serve immideately. Enjoy!