Roasting enhances walnuts in every way. Heat gets the natural oils in walnuts moving and aromatizing, shaking and baking -- the culinary equivalent of turning the volume up to "11." You want to roast walnuts whole if possible; roughly chopped walnut pieces don't cook evenly because of the differences in size. Oil is optional, but it amplifies the taste sensations and aromatic qualities you get from roasting and creates a crispness you can't get with dry-roasting.
Heat the oven to 350 degrees Fahrenheit. Shell the walnuts, if necessary.
Sift through the walnuts and discard any pieces of shell or discolored, soft nuts. Place the walnuts in an even layer on a sheet pan or cookie sheet.
Coat the walnuts with a thin layer of walnut oil or a neutral vegetable oil, such as canola, if you're roasting them for out-of-hand eating. Oiled walnuts develop a crisper texture than dry walnuts and a more complex flavor. However, if you're using the walnuts for a baking recipe -- muffins, for instance -- the extra oil will skew the fat-flour ratio, so you should roast them dry.
Roast the walnuts for another 5 minutes and take them out of the oven. The walnuts should have smoky aroma and a dark brown color about three shades darker than raw walnuts.
Transfer the walnuts to a cool pan or dish and spread them out in an even layer. Season the walnuts to taste with kosher salt while they're still hot, if desired. Mix the walnuts on the pan to distribute the salt.
Let the walnuts cool to room temperature. Store roasted walnuts in an airtight container in the refrigerator for up to one year.
Sift through the walnuts and discard any pieces of shell or discolored, soft nuts. Lay the walnuts in an even layer on a sheet pan or cookie sheet.
Coat whole walnuts with a thin layer of oil, if desired. If you're roasting walnut pieces, don't oil them; oil will make it more difficult to roast them evenly.
Place the walnuts in an even layer in a saute pan and set them on a large burner. Set the heat to medium.
Heat the walnuts until their aroma wafts about the stove, which indicates one side has started roasting, and they need a quick stir. Thoroughly stir the walnuts and let them roast undisturbed for another 3 or 4 minutes.
Taste the walnuts and check for a crisp texture and smoky flavor. Pan-roasted whole walnuts need about 10 to 12 minutes' total cooking time; walnut pieces roast in about 8 to 10 minutes.