Preheat the oven to 180 degrees C. Spread the nuts out on a small baking pan and toast lightly, about 5 minutes. Set aside to cool.
In the food processor, blend ¼ cup of the walnuts and the garlic; add the broccoli, basil, broth, nutmeg, ¾ teaspoon salt and the lemon juice, and pulse until well combined. Slowly drizzle in the oil through the feed tube with the processor running and puree until a slightly chunky sauce forms, scraping down the sides of the bowl as needed. Add 2 tablespoons of the cheese and pulse until incorporated.
Cook the pasta according to the package instructions. Drain, reserving 1/2 cup of the pasta water.
In a mixing bowl, toss the pasta with the pesto, along with some of the reserved cooking liquid. Add the remaining cooking liquid if needed and additional salt to taste.
Transfer the pasta to serving dish and sprinkle with the remaining cheese. Chop the remaining walnuts and scatter on top. Garnish with basil leaves and serve.