Preheat the oven to 350°. In an enameled cast-iron casserole, heat the oil. Season the beef with salt and pepper and cook over high heat until browned on 2 sides, about 7 minutes total. Transfer the meat to a bowl and pour off all but 1 tablespoon of the fat.
Step 2 Make the Braised Beef Filling
Add the onion, carrot, garlic and a large pinch of salt to the casserole. Cook over moderately low heat until the vegetables soften, 5 minutes. Stir in the tomato paste until shiny, 10 seconds. Stir in the wine; boil over moderately high heat until reduced by half, 4 minutes. Add the stock and thyme and bring to a boil. Add the beef and any accumulated juices. Cover and braise in the oven until the meat is tender, 1 hour. Discard the thyme.
Step 3 Make the Braised Beef Filling
Transfer the beef and vegetables to a bowl. Reduce the braising liquid over high heat to 1/2 cup, 5 minutes; let cool. Coarsely shred the beef and mix well with the vegetables. Stir in the ricotta, Parmesan and parsley and season with salt and pepper. Stir in the egg and cool braising liquid. Cover and refrigerate.
Step 4 Make the Pasta Rectangles
In a food processor, pulse the flour and salt. With the machine on, add the eggs; process until the dough looks like wet sand. On a work surface, knead the dough until smooth. Wrap in plastic and let stand for 15 minutes.
Step 5 Make the Pasta Rectangles
Cut the dough in half and flatten each piece to a 1/4-inch thickness; keep one piece covered. Run the other piece through progressively narrower settings on a pasta machine, through the thinnest setting. Cut the sheet into nine 6-by-5 1/2-inch rectangles. Repeat with the second piece of dough.
Step 6 Make the Pasta Rectangles
Cook the pasta in a pot of boiling salted water until al dente, about 1 minute. Transfer the pasta to a bowl of cold water. Drain and pat dry with paper towels; coat lightly with oil.
Step 7 Make the Bechamel Sauce
In a saucepan, melt the butter. Add the flour and whisk over moderately high heat to a thick paste, about 20 seconds. Gradually whisk in the milk and bring to a boil, whisking constantly until the sauce thickens. Reduce the heat to low and simmer, whisking, until no floury taste remains, 7 minutes. Stir in the cheese; season with nutmeg, salt and pepper.
Step 8 Finish the Dish
Preheat the oven to 350°. Butter a 9-by-13-inch baking dish. On a work surface, arrange the Pasta Rectangles with the shorter end facing you. Fill each rectangle with about 3 tablespoons of the Braised Beef Filling and loosely roll up lengthwise. In the prepared baking dish, spread 1 cup of the Béchamel Sauce in an even layer. Arrange the cannelloni, seam side down, in the dish. Pour the remaining Béchamel Sauce over the cannelloni and sprinkle with the grated cheese.
Step 9 Finish the Dish
Cover the cannelloni with foil and bake for about 40 minutes, until hot and bubbling. Remove from the oven and uncover. Preheat the broiler. Broil about 6 inches from the heat until golden brown, about 3 minutes. Let rest for 15 minutes.
Step 10 Finish the Dish
In a medium skillet, melt the butter. Add the sage leaves and cook over moderate heat until crisp, about 1 minute per side. Using a slotted spoon, transfer the sage to a paper towel–lined plate. Add the toasted walnuts and lemon juice to the pan and swirl to blend. Spoon the topping over the cannelloni, sprinkle with the fried sage and serve hot.
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