Preheat the oven to 300 degrees F. Grease two cookie sheets and set aside.
Whisk egg white in a small bowl. Place chopped walnuts in another small bowl.
Cream butter, brown sugar, and egg yolk in a mixing bowl until light and fluffy, about 2 minutes.
Add flour, vanilla, and salt; mix until well combined.
Scoop dough into 1 1/2-inch balls. Dip in egg white, then roll in walnuts until coated. Place 2 inches apart on the prepared cookie sheets.
Bake in the preheated oven until slightly puffed, about 5 minutes.
Remove cookies from the oven. Use your thumb to gently press an indent in the center of each cookie. Spoon jam into each thumbprint, filling it to the brim.
Return cookies to the oven and bake until set, about 8 minutes.
Remove from the oven and transfer to wire racks to cool completely.