Cookies: In large bowl, beat butter with sugar until fluffy; beat in egg, fancy molasses and cooking molasses. In separate bowl, whisk together flour, ginger, cinnamon, cloves, baking soda and salt; stir into butter mixture just until combined.
Roll by 1 tbsp into balls. Arrange, 2 inches apart, on parchment paper–lined rimless baking sheets. Supporting opposing sides of 1 ball with 2 fingers, press opposite index finger into centre to create well 1/2 inch deep and 3/4 inch wide. Pinch together any cracks around edge. Repeat with remaining balls. Refrigerate until firm, about 30 minutes.
Bake, 1 sheet at a time, in 350°F oven until no longer shiny, about 12 minutes. Remove from oven; using rounded back of 1 tsp measure, gently press wells to reshape. Let cool on pans for 5 minutes; transfer directly to racks to cool completely.
Spiced Cream Cheese Icing: While cookies are cooling, in bowl, beat cream cheese with butter until fluffy; beat in molasses, vanilla, cinnamon, ginger and cloves. Beat in icing sugar until smooth. Spoon or pipe heaping 1 tsp of the icing into well of each cookie.
Garnish: Sprinkle icing with ginger. Refrigerate until icing is set, about 30 minutes. (Make-ahead: Refrigerate in airtight container for up to 3 days.)
Makes about 40 cookies.