The day before you plan on cooking your steak, place the rib eye in a resealable baggie with the rosemary, garlic, thyme, and olive oil. Move the herbs and garlic around the bag with the steak to coat the steak with the flavors. Store in the refrigerator over night.
Preheat a grill pan to medium high heat. Preheat the oven to 400 degrees. Remove the steak from the refrigerator 30 minutes before cooking.
Season the steak evenly with 1 teaspoon salt and place it directly on to the preheated grill. Do not move it. Allow the steak to sear for 5 minutes to form a golden brown crust. Flip the steak and repeat the same method on the other side. Remove the steak to a baking sheet and place in the oven for 10 minutes for medium doneness. Set aside and allow the steak to rest for 15 minutes before slicing.
Meanwhile, in a small bowl mix together the lemon zest, lemon juice, 2 teaspoons olive oil, thyme, rosemary, 1/8 teaspoon salt and 1/8 teaspoon pepper. Set aside. Heat a small non stick saute pan over medium heat. Add 1 teaspoon of olive oil and slide in the egg being careful not to break the yolk. Season with remaining 1/8 teaspoon salt and pepper. Reduce the heat to medium low and cook the egg gently until the white is set and the yolk is still soft, about 3 minutes.
Place the rested steak on a serving plate. Spoon 1 teaspoon herb oil directly over the steak. Place the sunny side up egg on top of the steak and scatter the arugula leaves organically over the egg. Drizzle with the remaining herb oil and top with a few grinds of freshly ground black pepper.