Season the steak: Pat the steak dry.
Season both sides with 1–1.5 teaspoons kosher salt and 1/2 teaspoon black pepper. Let it sit at room temperature for 20–30 minutes while you prep everything else.
Cook the pasta: Bring a large pot of salted water to a boil. Cook pasta until just al dente.
Reserve 1/4 cup pasta water, then drain and spread on a sheet pan to cool slightly.
Make the dressing base: In a large bowl, whisk olive oil, lemon zest, lemon juice, red wine vinegar, Dijon, a pinch of red pepper flakes, and a small pinch of salt and pepper.
Prep the vegetables: Halve cherry tomatoes, dice cucumber, and slice red onion thinly. If the onion is sharp, soak slices in cold water for 10 minutes and drain.
Sear the steak: Heat a large skillet or grill pan over medium-high. Add a drizzle of oil.
Sear steak 3–4 minutes per side for medium-rare (thickness varies). Transfer to a plate and rest 5–10 minutes.
Make garlic-Parmesan butter: Lower the skillet heat to medium. Add butter.
When melted and foamy, add minced garlic. Cook 30–45 seconds until fragrant but not browned. Remove from heat.
Stir in half the Parmesan and 1–2 tablespoons reserved pasta water to make a loose, glossy sauce.
Toss the pasta: Add warm pasta to the large bowl with the dressing. Toss to coat. Pour in the garlic-Parmesan butter and toss again until the pasta is well-glossed.
Add vegetables and herbs: Fold in tomatoes, cucumber, red onion, arugula, parsley, and chives.
Sprinkle in most of the remaining Parmesan, saving a little for garnish.
Slice the steak: Slice against the grain into thin strips or bite-size cubes. Add to the pasta salad and gently toss. Taste and adjust with more lemon, salt, or pepper as needed.
Finish and serve: Top with reserved Parmesan and an extra drizzle of olive oil.
Serve slightly warm or at room temperature for the best flavor.