Put bones, onions, carrots, celery, oil, salt, and 10 cups cold water into a medium pot and bring to a boil over high heat, skimming any foam that rises to the surface. Reduce heat to medium and simmer until stock has reduced by half, about 1 hour. Strain stock into a medium bowl, discarding solids. Strain stock again through a large sieve lined with a double layer of cheesecloth into a small saucepan. Keep stock hot over low heat.
To assemble the lasagne, use 1 sheet of the music-paper bread (1) to line the bottom of a large wide deep dish, breaking it into pieces, if necessary, to fit inside the dish and completely cover its bottom. Scatter a few pinches of cheese over bread. Moisten bread here and there with a small ladleful (about 1⁄4 cup) of the hot stock (2). Spread a larger ladleful (about 1 cup) of the hot tomato sauce completely over bread (3), then sprinkle with a small handful (3–4 tbsp.) of cheese (4). Repeat layering process in this order 5 more times, gently pressing bread onto tomato sauce before adding cheese and stock, making 6 layers in all and ending with tomato sauce and cheese. Set aside until bread is completely hydrated and tender, about 30 minutes.
Bring remaining stock (you should have about 3 cups) to a simmer in a medium skillet over medium heat. Working in two batches, crack 3-4 eggs into simmering stock, cover skillet, and poach until whites are just firm, 2 1⁄2-3 minutes per batch. Transfer eggs with a slotted spoon to paper towels to drain. Set 1 poached egg in center of lasagne. Serve lasagne at room temperature. Topping each serving with a poached egg.