Heat oven to 350°F. In small bowl, stir glaze ingredients until smooth and thin enough to drizzle. Cover; set aside.
In large bowl, beat 1/2 cup butter, the shortening, 1/2 cup granulated sugar and the brown sugar with electric mixer on medium speed until thoroughly mixed. Beat in eggs and 1 teaspoon vanilla until well blended. Beat in oats, cinnamon, nutmeg, baking soda and flour, scraping side and bottom of bowl once, just until blended. Stir in apples and walnuts.
On ungreased cookie sheets, drop dough by rounded tablespoonfuls (scant 1/4 cup) about 2 inches apart.
Bake 13 to 17 minutes or until edges are golden brown and centers are set. Cool 2 minutes; remove from cookie sheets to cooling racks. Using spoon, drizzle about 1 teaspoon glaze onto each warm cookie. Cool completely, about 10 minutes.