Preheat the oven to 425°F (220°C). Toss the shoestring fries in a bowl with olive oil, salt, pepper, and lemon zest. Spread them out in a single layer on a baking sheet and bake for 20-25 minutes, turning halfway through, until golden and crispy. Remove from the oven and immediately sprinkle with Parmesan cheese. Set aside.

Preheat a large oven-safe skillet over medium-high heat. Season the strip steaks generously with salt and pepper. Add 2 tablespoons of butter to the skillet. Once melted and bubbling, add the steaks. Sear for 3-4 minutes on each side until a nice crust forms. Add the chopped basil and the remaining butter to the skillet. Spoon the melted basil butter over the steaks as they cook for an additional minute.

Transfer the skillet to the preheated oven and roast for 5-7 minutes, or until the desired doneness (135°F for medium-rare). Remove from the oven and let rest for 5 minutes.

In the same skillet, remove excess fat, leaving about a tablespoon. Add the beef broth and Worcestershire sauce, scraping up any browned bits from the bottom of the pan. Bring to a simmer and reduce for 5-10 minutes until slightly thickened. If a thicker sauce is desired, mix cornstarch with a little water to create a slurry and add it to the sauce, stirring until thickened.

Slice the strip steaks and plate alongside the lemon-Parmesan shoestring fries. Drizzle the velvety brown sauce over the steak and garnish with fresh parsley.