Heat oven to 350. Grease 2 large heavy cookie sheets.
Mix first 4 ingredients together in a small bowl. Using electric mixer, beat butter, sugar and vanilla in a large bowl until light and fluffy.
Beat in egg. Stir in dry ingredients, then vanilla milk chips and nuts. Drop cookie dough by scant 1/4 cup onto prepared cookie sheets, spacing evenly. Bake until golden brown, about 15 minutes. Cool cookies on sheets for 5 minutes. Transfer cookies to rack to cool.