Step 1. Bring a large pot of salted water to a boil. Add pasta and cook according to package directions until al dente. Reserve ½ cup pasta water, then drain pasta and toss with olive oil to prevent sticking. Set aside.
Step 2. Melt butter in a small sauté pan over medium heat (it’s helpful to use a light-colored pan so you can see the butter turn color). Stir constantly until butter begins to darken. When butter reaches a caramel hue, remove from heat and transfer to a heatproof measuring cup, leaving any browned bits behind.
Step 3. Set pasta pot over medium heat and add 4 tablespoons brown butter. Return pasta to pot and toss to coat. Drizzle in about half the pasta water and toss to coat, reducing heat to medium-low if pasta starts sticking.
Step 4. Continue tossing and stirring, drizzling in water as needed to make a shiny, emulsified sauce. Add walnuts and toss to combine. When sauce has emulsified and walnuts are clinging to the pasta, add half the cheese and stir to combine. Taste and add salt as needed.
Step 5. Divide pasta among serving bowls (you may have some left over). Drizzle remaining brown butter over the top of the pasta, followed by remaining Goat Cheese, bread crumbs, chives, and a few grinds of black pepper. Serve.