Heat butter in a sauté pan over medium-high heat. Cook until melted and beginning to turn golden brown.
Remove from heat and add molasses, fresh ginger, ground cinnamon, ground ginger, ground nutmeg, ground cloves, heavy cream, and vanilla.
Pour brown butter mixture into a heat-proof bowl and chill in the freezer for 15-30 minutes, stirring occasionally, until butter is the texture of a regular stick of butter left out at room temperature.
Transfer butter to the bowl of a stand mixer. Add brown sugar and mix on medium speed until light and fluffy, about 5 minutes.
Turn mixer speed down to low and add the salt and flour. Mix just until combined.
Lightly flour a work surface and roll your dough into a large rectangle until it's 3/4" thick.
Cut out 12-16 gingerbread shortbreads with a gingerbread man cookie cutter. Transfer to a parchment-lined baking sheet.
Preheat oven to 350°F and place sheet tray of cookies in the freezer for 10 minutes until firm.
Bake for 16-18 minutes until slightly golden brown and firm. Let chill and eat plain or coated in royal icing.