Combine the butter, golden syrup, molasses, and granulated sugar in a saucepan over medium heat. Continue heating until the mixture boils.
Remove the mixture from the heat, and allow it to cool. Add the mixture to the bowl of a stand mixer. Add the egg and mix.
In a separate bowl, whisk together the flour, baking soda, ginger, and cinnamon. Add the dry ingredients to the egg mixture and mix until smooth.
Let the dough rest in the refrigerator overnight.
Preheat the oven to 350F. In order to minimize the incorporation of excess flour into the dough, place a sheet of parchment paper above and below the ball of dough. Use a rolling pin to roll the dough to about ¼-inch thickness. Using a fluted round cookie cutter or other cookie cutter of your choice, cut the cookies and place on a baking sheet. Bake for 7-8 minutes, or until the cookies start to slightly darken. Allow the cookies to cool.
Prepare the buttercream by whisking together butter, powdered sugar, and vanilla extract until smooth. Use about 1-2 tsp of buttercream and two cookies to make each of the mömmukökur.