Peel and finely chop 1 garlic clove.
Heat oil in a large Dutch oven or other wide heavy pot over medium-high heat.
Add celery, carrot, onion, chopped garlic and generous pinch salt and pepper.
Reduce heat to medium and cook, stirring occasionally, until vegetables are softened, about 5 minutes.
Add rabbit pieces and broth. Tuck pieces into broth to mostly cover and bring to a gentle simmer.
Reduce heat to low, cover and cook, turning pieces occasionally, until meat is tender, at least an hour.
Meanwhile, wipe mushrooms with paper towels to remove any dirt or grit.
Keeping stems attached, trim any discoloration from stems.
Cut mushrooms into 1/4-inch dice.
Gently smash and peel remaining garlic clove.
In a medium skillet, heat garlic and 1 tablespoon butter over medium heat. When butter is melted and bubbling, add mushrooms; cook, stirring frequently, for 3 minutes.
Stir in parsley and pinch salt. Remove from heat.
Remove rabbit from heat. Transfer rabbit pieces to a plate, reserving sauce in pan. When cool enough to handle, pull meat from bones, taking care to remove and discard any small shards of bone.
Strain sauce and return to pan.
Chop meat into small pieces and return to sauce.
Add mushroom mixture to sauce, bring to a gentle simmer and cook for at least 30 minutes, until the ragù starts to thicken noticeably; remove from heat and adjust seasoning.
Bring a large pot of salted water to boil.
Just before adding pasta to water, gently reheat sauce; add remaining 2 tablespoons butter to sauce and stir to combine.
Add pasta to water and cook until just tender, about 2 1/2 minutes (fresh pasta cooks much faster than dried).
Drain pasta and transfer to a large serving bowl; add ragù and toss well to combine.
Serve immediately, sprinkled with cheese.
