Bring a large pot of heavily salted water to a boil. Coarsely chop 1 packed cup fresh herb leaves. Finely grate 1 ounce Parmesan cheese (about 1/2 packed cup) or measure out 1/3 cup store-bought grated.
Add the pasta to the boiling water and cook according to package instructions for 1 minute less than al dente. Meanwhile, start the sauce.
Cut 8 tablespoons unsalted butter into a few pieces and place in a large skillet. Melt over medium heat. Add 1 tablespoon za’atar and continue cooking, swirling the pan occasionally and scraping up any browned bits at the bottom of the pan with a wooden spoon, until the butter has a nutty, fragrant aroma and is a toasty-brown color, 2 to 3 minutes. Remove the pan from the heat. Carefully squeeze the juice from 1/2 medium lemon (about 1 1/2 tablespoons) into the pan (it will spatter). Add 1 teaspoon kosher salt and stir to combine.
When the pasta is ready, reserve 2 cups of the cooking water. Drain the pasta.
Pour the reserved pasta water into the sauce and bring to a simmer over medium-high heat. Simmer, scraping up any browned bits at the bottom of the pan with a wooden spoon, until the liquid reduces by half, about 3 minutes. Add the pasta and toss continuously until the sauce thickens a bit more and coats the pasta, and the pasta is al dente, about 1 minute more.
Remove the pan from the heat. Add the herbs and half of the Parmesan and toss to combine. Serve garnished with the remaining Parmesan and a generous sprinkle of za’atar.