Preheat oven to 375 degrees with racks in the lower and middle positions. Finely chop 1 1/2 cups walnut pieces.
Beat butter and sugar until pale and fluffy. Beat in egg and rum. Stir in flour and salt, scraping down sides of bowl. Mix in chopped walnuts.
Drop dough (about 2 teaspoons per cookie) onto parchment-lined baking sheets, spacing about 3 inches apart. Bake for 4 minutes; remove from oven, and top with remaining 3/4 cup walnut pieces. Rotate sheets, and return to oven. Bake until golden brown around edges, 8 minutes more. Transfer cookies on parchment to wire racks. Let cool slightly, about 10 minutes.
Cookies can be stored at room temperature for up to 3 days.