1. In a bowl, with an electric mixer on high speed, beat butter, brown sugar, egg yolk, and vanilla until smooth. Stir or beat in flour and oats until well blended.
2. In a small bowl, beat egg white and 2 teaspoons water to blend. Put walnuts in a shallow bowl. Shape dough into 2-teaspoon-size balls and dip in egg white mixture; shake off excess, then roll in walnuts. Place 1 inch apart on buttered or cooking parchment-lined 12- by 15-inch baking sheets. Press your thumb into the center of each cookie to make a 1/2-inch-deep indentation.
3. Bake cookies in a 300° oven until lightly browned, about 15 minutes; if baking more than one pan at a time, switch pan positions halfway through baking.
4. Carefully fill each indentation with about 1 teaspoon jam (if indentations have disappeared, make them again with the rounded end of a wooden spoon). Return to oven and bake until cookies are golden brown and jam has melted, about 15 minutes longer. With a wide spatula, transfer cookies to racks to cool completely.
Note: Nutritional analysis is per cookie.