Mix all brine ingredients in a large stock pot and heat until salt and sugar are dissolved. Allow brine to cool. Poke holes in the squirrels to better absorb the marinade Add raw squirrels and ensure they are completely covered by brine solution. Refrigerate for 24-48 hours.
Once squirrels are brined, remove them from brine and rinse. Pat dry with paper towels.
Heat oil over medium high in a Dutch oven, or other heavy lidded pan, until oil shimmers. Sear squirrels in batches until all sides have a nice brown sear. Take your time and don’t crowd the pan. Two squirrels at a time are usually plenty.
When searing is finished, remove squirrels from the pan and add onions, garlic, and thyme sprigs to the pan. Cook about five minutes, until onions wilt. Add squirrels back to the pan. Add white wine, butter, and salt. Cook for thirty minutes on low, then add potatoes and olives. Cook an additional 30-60 minutes, stop cooking when squirrel meat shreds easily. Remove squirrels from the pan and shred meat off the bones. Add meat back to potatoes, olives, and broth. Add more salt if needed.
Cook pasta according to package directions. Once pasta is cooked, before you drain water, add 1/2 cup of the cloudy pasta water to the squirrels, potatoes, and olives. Add the drained pasta into the pan with shredded squirrels and cook over low to let juices absorb into pasta. Do not rinse pasta after draining, add directly to squirrels. The carbohydrates in the pasta water make the sauce silky and rich.
Be sure to drink the same wine you used to cook! Top with fresh chopped parsley and a squeeze of lemon juice if you have it.