To a large pot of salted, boiling water add spaghetti and cook until al dente. Reserve 1/3 cup cooking liquid and drain pasta in a colander. Return pasta to kettle with reserved liquid.
While pasta is cooking, in a large heavy skillet heat oil over moderate heat until hot but not smoking and cook broccoli with salt, to taste, stirring occasionally, until crisp-tender, about 5 minutes. Add garlic and cook, stirring, until golden, about 30 seconds.
To pasta add broccoli mixture, Brie, and walnuts, tossing until cheese is just melted, and season with salt and pepper.