PREHEAT oven to 375° F.
MELT morsels in small saucepan over low heat for 3 to 5 minutes, stirring constantly, until smooth. Remove from heat and cool slightly.
COMBINE melted morsels, flour, 1 cup butter, brown sugar, granulated sugar, egg, half the milk and 2 teaspoons vanilla extract in large mixing bowl. Beat with an electric mixer on low speed, scraping the bowl often, until well mixed, 1 to 2 minutes. Divide dough in half and cover each in plastic wrap. Refrigerate for about 1 hour or until firm.
ROLL out dough on lightly floured surface, one half at a time, to 1/8-inch thickness. Cut with cookie cutters and place 1-inch apart on baking sheets.
BAKE 5 to 8 minutes, or until edges are lightly browned. Cool on baking sheets for 2 minutes; remove to wire racks to cool completely.
MIX powdered sugar, remaining butter, milk and vanilla extract in small mixing bowl until fluffy, scraping the bowl often, about 2 minutes. When cookies are cool, spread on frosting.