Whisk together first 3 ingredients in a small bowl; set aside.
Combine flour, salt, ginger, and cardamom; set aside.
Beat 1/2 cup butter and cream cheese at medium speed with an electric mixer until fluffy. Add granulated sugar and 1/4 cup brown sugar; beat until creamy. Add egg, and beat until blended. Slowly add buttermilk mixture to butter mixture; beat 2 minutes. Stir flour mixture into butter mixture just until combined. Fold in crystallized ginger and chocolate morsels. Pour mixture into a lightly greased 9-inch square pan.
Bake at 350° for 53 minutes or until cake springs back when lightly touched in center. Cool completely in pan on a wire rack.
Melt 1/4 cup butter in a large skillet over medium heat. Add 1/4 cup brown sugar, stirring until sugar dissolves. Add pears; cook 5 minutes or until slightly thickened, stirring gently to coat. Serve warm sauce over cake.