In a bowl, with a mixer on medium speed, beat butter, brown sugar, and molasses until blended. Beat in egg and vanilla until smooth, scraping down sides of bowl as needed.
In another bowl, mix flour, baking soda, and salt. Stir into butter mixture until well blended. Stir in walnuts.
Drop dough in 1 1/2-teaspoon portions, 1 1/2 inches apart, onto buttered or cooking parchment-lined 12- by 15-inch baking sheets.
Bake cookies in a 350 oven until slightly darker brown at edges, 10 to 12 minutes; if baking more than one pan at a time, switch pan positions halfway through baking.
With a wide spatula, transfer cookies to racks to cool. If hot cookies start to break, slide a thin spatula under them to release; let stand on pan to firm up, 2 to 5 minutes, then transfer to racks to cool completely.
