Parsley-and-cheese sausage is available at many butcher shops, but you can also substitute any sausage you like.
Heat oven to 350 degrees. Place a large saute pan over medium-low heat. When hot, add olive oil and sausage, pricked every few inches with the point of a knife, and cook until nicely browned, about 10 minutes. Turn sausage over, prick with knife, and cook until browned, about 10 minutes.
Transfer sausage to a baking pan, and place in oven, leaving rendered fat in saute pan. Add onions to saute pan, and stir. Add 1/2 cup water, and scrape brown bits from bottom of pan with a wooden spoon. Cover, and cook over medium heat, stirring occasionally, until onions are very soft, 15 to 20 minutes, adding a second 1/2 cup water if pan becomes dry. Remove lid, and stir in thyme. Season with salt and pepper. Remove sausage from oven, and serve.