Tagliatelle with Chestnuts, Pancetta, and Sage

1 tablespoon extra-virgin olive oil

1 small onion , finely chopped

4 garlic cloves, minced

2 tablespoons finely chopped fresh sage

8 ounces bottled peeled roasted whole chestnuts , coarsely crumbled (1 1/2 cups)

8 ounces dried flat egg pasta such as tagliatelle or fettuccine

2 ounces finely grated Parmigiano -Reggiano (1 cup)

2 tablespoons unsalted butter

1 tablespoon finely chopped fresh flat-leaf parsley