Preheat the oven to 500°. Spread the walnuts on a baking sheet and toast in the oven for about 6 minutes, or until golden brown. Transfer to a plate and let cool.
In a small bowl, combine the shallot, garlic, anchovies and their oil, lemon juice, mustard, salt, and pepper. Using a fork, whisk in the olive oil until smooth.
Just before serving, combine the romaine and arugula in a large salad bowl. Pour the dressing over the greens and toss well. Add the toasted nuts and the Parmesan. Toss again and serve.