Preheat oven to 350 degrees. Sift flour, cinnamon, baking soda, baking powder, and 1/4 teaspoon salt into a bowl. Beat 1 stick butter and the sugars until pale and fluffy. Beat in egg and vanilla.
Beat flour mixture into butter mixture. Mix in zucchini, oats, and walnuts. Refrigerate until firm, about 1 hour.
Using a 1 1/2-inch ice cream scoop (about 2 tablespoons), drop dough onto parchment-lined baking sheets, spacing about 2 inches apart. Bake until edges are golden, about 17 minutes. Let cool on a wire rack.
Beat together remaining 1/2 stick butter, the cream cheese, and confectioners' sugar until smooth. Spread 1 heaping tablespoon filling onto the flat side of 1 cookie, and sandwich with another cookie. Repeat with remaining filling and cookies.