Preheat oven to 375°F. Line baking sheets with parchment paper or silicone baking mats.
In a large bowl using an electric mixer, beat butter, brown sugar, and granulated sugar on medium-high speed until light and creamy, about 3 minutes. Add egg, vanilla, and molasses, beating on high speed until combined, about 1 minute.
In a medium bowl, whisk together flour, baking soda, salt, cinnamon, cloves, and oats.
With mixer on low speed, slowly add dry ingredients to wet ingredients. The dough will be thick and sticky.
Scoop 2 heaping tablespoons of dough per cookie and place 3 inches apart on baking sheets. Bake for 10-13 minutes until lightly golden around edges but very soft in centers.
Allow cookies to cool on baking sheet for 10 minutes before transferring to cooling rack to cool completely.
Beat butter on high speed until creamy, about 1 minute. Add confectioners' sugar and beat on medium speed for 1-2 minutes. Pour in heavy cream and vanilla, beating on high speed for 2-3 minutes. Add salt to taste.
Spread about 1½ tablespoons of cream filling on bottom side of half the cookies. Top with remaining cookies, right side up.