Cream shortening in a large mixing bowl; gradually add sugar, beating well. Add eggs, one at a time, beating well after each addition. Stir in vanilla.
Sift together flour, soda, and salt; add to creamed mixture alternately with milk, beginning and ending with flour mixture. Stir well after each addition. Stir in pecans. Cover and chill at least 1 hour.
Drop by teaspoonfuls 2 inches apart onto greased cookie sheets. Bake at 375° for 8 to 10 minutes. Remove from cookie sheets, and cool on wire racks. Spoon Burnt Butter Glaze over cookies. Cool completely.