(1 cookie sandwich)
Preheat oven to 350°F. For the Cookies, mix flour, baking soda, salt, cinnamon and nutmeg in large bowl. Set aside. Beat butter and sugars in large bowl with electric mixer on medium speed until light and fluffy. Add eggs and vanilla; mix well. Gradually beat in flour mixture on low speed until well mixed. Stir in oats and pecans.
Drop dough by rounded tablespoons or medium cookie scoop 2 inches apart onto greased or parchment paper-lined baking sheets.
Bake 12 to 14 minutes or until lightly browned. Cool on baking sheets 1 minute. Remove to wire racks; cool completely.
Meanwhile, for the Ginger Buttercream Filling, beat butter in large bowl with electric mixer on medium speed until light and fluffy. Gradually add confectioners’ sugar, beating until well blended after each addition. Add milk and vanilla; mix well. Add cinnamon and ginger; beat until well blended. To assemble cookie sandwiches, spread about 2 tablespoons of the filling on the flat side of 1 cookie. Top with a second cookie. Repeat with remaining cookies and filling.