Preheat the oven to 350 degrees F. Line half-sheet pans with parchment paper and set aside.
In a medium bowl, whisk together the flour, ginger, cinnamon, baking soda, salt, nutmeg, and cloves. Set aside.
In a large bowl using a hand mixer or a stand mixer fitted with a paddle attachment, beat the butter, brown sugar, and granulated sugar together on medium-high speed until light and creamy. Add the molasses, egg, and vanilla extract. Beat until well combined. Stop and scrape down the sides of the bowl as needed.
Add the dry ingredients and mix on low until just combined. Stir in the oats.
Use a cookie scoop to drop the dough onto the prepared baking sheet, about 1 heaping tablespoon of dough per cookie, leaving about 2 inches of space in between because the cookies will spread.
Bake for 10 to 12 minutes or until cookies are set around the edges but still soft in the center. Check early to be safe. Let the cookies cool for 5 minutes on the baking sheet. Transfer to a wire cooling rack and cool completely.
While the cookies are cooling, make the frosting. In a large bowl, beat the butter with a hand mixer (or a stand mixer with the paddle attachment) on high until creamy, about 1 minute. Add the confectioners’ sugar and mix on medium for 1–2 minutes. Add the heavy cream (or milk), vanilla, and pinch of salt. Beat on high for another 2 minutes until light and fluffy.
Spread about 1 heaping tablespoon of the cream filling onto the bottom side of half the cookies. You can pipe, spoon it on, or use a little cookie scoop.
Place the remaining cookies on top, right-side up, to make sandwiches. Enjoy!