Bring a large pot of salted water to a boil. Cook pasta according to package instructions. Drain cooked pasta.

In the same large pot over medium heat, melt the butter and cook an additional 1 to 2 minutes. If you want browned butter, cook until it has a light brown color. Add the garlic and thyme, stir, and cook until fragrant, about 1 minute. Be careful not to over-brown the butter or garlic.

Remove the pot from the heat. Add the olive oil, lemon juice, lemon zest, and salt. Stir to combine. Add the pasta, arugula, and pine nuts and toss to coat with the sauce. The arugula will wilt.

Transfer pasta to a serving bowl, platter, or individual plates. Top with shaved parmesan cheese. Serve immediately.