There are many fine restaurants with excellent, local food, but here's a thought out of the box: this is the season for church suppers. They're usually held at the church on Friday, Saturday, or Sunday evenings and feature traditional, local foods (chicken pie, boiled dinner), always prepared by, and often raised/produced by the church's members--vegetables from the gardens and dairy from the farms. I've had many a fantastic meal with some very hospitable, local people with great conversations on topics you wouldn't be likely to hear in town and yes, no religion (except perhaps a brief, ecumenical grace at the start). And of course, homemade pies for dessert!
For the best way to find and reserve a place at one, check the local calendar of local events, e.g., the Burlington-based Seven Days: