Heat 3 tablespoons of the olive oil in a large, deep, ovenproof pot over medium heat until hot.
Season the shanks with salt and pepper and brown on all sides, about 10 minutes.
Remove to a plate.
Raise the heat to medium-high, add the mushrooms, and do not move them until they begin to brown, about 1 minute.
Season with salt and pepper and sauté until brown all over, about 5 minutes.
Remove to another plate and reserve for final assembly.
Reduce the heat to medium, add the remaining 3 tablespoons olive oil to the pot, and heat until hot.
Add the garlic and sauté briefly until light brown.
Add the onion, carrot, and celery, season with salt and pepper, and sauté until light brown, about 8 minutes.
Add the wine and bay leaf, bring to a boil over high heat, and cook until reduced by half.
Add the stock and tomatoes and bring to a boil again.
Season with salt and pepper.
Return the meat to the pot, cover, and place in the oven to braise until fork tender.
Test at 2 hours, but the shanks may take as long as 4 hours.
Let the meat cool in the liquid to room temperature.
Remove from the braising liquid and reserve separately.
Skim off and discard the fat from the braising liquids.
(The recipe may be made to this point a day ahead, covered, and refrigerated. See Chef's Note.) Bring a large pot of water to a boil.
Add salt and the pasta and cook until al dente, about 12 minutes.
Meanwhile, pour the defatted braising liquids into a saucepan and bring to a boil over high heat.
Reduce the heat and simmer for about 10 minutes, skimming the surface all the while.
Add the basil and oregano.
Return the shanks to the sauce and simmer gently just until heated through, then remove and keep warm.
Add the mushrooms and pasta to the sauce and heat gently until warm through.
Pour onto a large, deep platter or divide among plates and top with the shanks.
Variation: If you wish to serve the meat off the bone (which turns this into a pasta dish rather than meat on a bed of pasta), tear the meat into big chunks and return it to the sauce with the mushrooms.
Cook just until reheated.
Add the pasta, stir, and heat through.
Taste for seasoning and serve immediately.
Chef's Note: It is acceptable and even preferable to do this dish a day ahead.
It is much easier to defat the braising liquid after it has been refrigerated.
The meat can even cook while you sleep.
For example, from midnight to 6:00 a.
You can also cook the whole dish in a covered pot on the stove-top.
There is a caramelization of flavors in oven braising that stove-top cooking does not replicate.
If the shanks are cracked cut through the bone in the middle they will cook through to tenderness more rapidly, closer to 2 hours than 4 hours.