In a small bowl, combine 2 tablespoons grated Parmesan cheese and 1 1/2 tablespoons room-temperature butter; mix until smooth and set aside.
Heat 1 teaspoon olive oil in a medium skillet over medium-high heat until shimmering.
Add the 12-ounce rib‑eye steak to the skillet and cook to desired doneness, about 4 minutes per side for medium-rare. Flip once using tongs.
Transfer the steak to a plate and let rest for about 5 minutes.
While the steak rests, return the skillet to medium-high heat if it has cooled. Add 1/4 cup balsamic vinegar, 1/4 cup finely chopped shallots, and 1/2 teaspoon (packed) dark brown sugar to the skillet. Bring to a boil and cook, stirring constantly, until the liquid reduces to a glossy glaze, about 1 minute. Remove from heat.
Divide 4 cups (lightly packed) arugula or mixed greens and the Parmesan cheese shavings between 2 plates. Squeeze one large lemon wedge over each plate.
Slice the rested steak against the grain and place the slices atop the arugula on each plate.
Top the steak slices with portions of the Parmesan butter so it melts, then drizzle lightly with the balsamic glaze.