Heat a medium or large (preferably cast iron) skillet on medium heat. Add a thin layer of olive oil. Note that the skillet needs to be big enough to hold the sauce and the squirrels.
Dice six tomatoes (set aside two) and lightly dust with kosher salt, freshly ground black pepper, and dry oregano. Cook until slightly soft, then add to food processor but do not blend yet.
In the same skillet, after tomatoes are cooked and removed, add a little more olive oil, then the finely diced onion. Lightly salt and pepper and sauté until brown, then add to food processor.
Once again, in same skillet, add a little bit of olive oil, followed by the squirrels. Lightly season with salt and pepper. Brown on all sides, then set aside.
Add remaining sauce ingredients (aside from two on-the-vine tomatoes) to food processor and blend thoroughly. Add sauce to the skillet and set to simmer.
Allow sauce to simmer for 10 minutes before adding squirrels to the sauce.
Braise the squirrels in the sauce. After 2 to 2-1/2 hours, the meat on squirrels will start to tenderize. At this point, you can remove the squirrels and carefully pick the meat from the bones. Add the meat back into the sauce.
Dice the final two tomatoes and add to the sauce, along with the rinsed and sliced baby portabella mushrooms. Allow all of this to simmer for 30 minutes longer.
In a separate large pot, boil pasta for 8-9 minutes until slightly al dente, then drain.
Salt the sauce to taste and, when satisfied, serve over pasta. Garnish with fresh basil, cut chiffonade, if you wish.