Preheat oven to 375 degrees F.

In a heavy saucepan over medium-low heat, melt butter. Whisk in flour and cook, whisking, 3 minutes. Whisk in milk and bring the sauce to a simmer. Simmer sauce, stirring, about 8 minutes. In a bowl, whisk eggs and add 1 cup sauce in a steady stream, whisking constantly to keep the eggs from cooking. Whisk egg mixture into remaining sauce and stir in dill, nutmeg, and salt and pepper. Keep dill sauce warm and its surface covered with plastic wrap.

In a pot of boiling salted water, cook penne until al dente. Rinse penne in colander and drain well. In a large bowl, stir together pot roast meat, beef broth, and gravy. In a buttered shallow 7 by-11-inch baking dish, place half of pasta evenly over the bottom. Top with 1/3 of the reserved sauce and then spread beef mixture evenly on top of the sauce. Top evenly with remaining pasta and pour the remaining sauce over the pasta. (Pour the sauce slowly to keep from overflowing and to allow it to sink in.) Bake in middle of oven until set, about 40 minutes. Sprinkle cheese over the pastitsio during the last 20 minutes of baking.