Heat oven to 375°F. In large bowl, beat brown sugar, crystallized ginger, 3/4 cup softened butter, the pumpkin and egg with electric mixer on medium speed until well blended. Stir in remaining cookie ingredients to form soft dough. (If dough is too soft, refrigerate about 15 minutes.)
Shape dough into 1-inch balls; place about 2 inches apart on ungreased cookie sheets. Slightly flatten balls.
Bake 7 to 10 minutes or until light golden brown. Immediately remove from cookie sheets to cooling racks. Cool completely, about 15 minutes.
In 2-quart saucepan, heat 1/3 cup butter over medium heat, stirring constantly, until light golden brown. Remove from heat. Stir in remaining frosting ingredients until smooth and spreadable. Immediately spread frosting on cooled cookies.