Preheat oven to 375°F.
Place the squirrel in a roasting pan and roast until golden brown, about 30 minutes. Set aside.
Heat the olive oil in a heavy-bottomed pot over medium-high heat. Add onion, carrots, garlic, herbs, and peppercorns. Cook 3–5 minutes, occasionally stirring.
Add the roasted squirrel to the pot. Pour in enough water (around 6 cups) to cover by about 2 inches. Bring to a gentle simmer and cook until the meat is fork-tender, about 1½ hours.
Remove squirrel and set aside to cool. Strain and reserve the cooking liquid; discard solids.
In a clean pot or skillet over medium heat, melt butter and whisk in flour. Cook the roux while stirring, until light golden brown, about 5 minutes.
Slowly whisk in the reserved cooking liquid a little bit at a time until your desired thickness is achieved. Simmer until slightly thickened, 10–15 minutes.
Pick the meat from the squirrel, discarding bones. Stir the meat into the gravy. Season with salt and pepper. Serve hot over biscuits, rice, mashed potatoes, or fried squirrel.