Preheat the oven to 375 degrees F. Line 2 baking sheets with parchment paper.
For the molasses cookies: Put the brown sugar, butter and vegetable shortening in a large bowl and beat with a hand mixer on medium speed until combined. Add the flour, molasses, eggs, cinnamon, ginger and baking soda and beat until combined.
Use a tablespoon to scoop the dough onto the baking sheets. Make sure there is ample space between each dough ball as the cookies will spread.
Bake until puffed on top, 10 to 12 minutes. Cool the cookies on the baking sheet on a rack for 5 minutes, then transfer the cookies to the rack to cool completely before frosting.
For the boiled frosting: Combine the granulated sugar, 3/4 cup water, cream of tartar and a pinch of salt in a medium heavy-bottomed pot, bring to a boil over medium heat and cook until a candy thermometer registers 245 degrees, about 3 minutes. Remove from the heat and fold in the marshmallows until melted.
Put the egg whites in the bowl of a stand mixer fitted with a whisk attachment and whip on medium-high speed into stiff peaks. Turn the mixer to low and slowly pour the hot sugar mixture into the egg whites. Beat on medium speed until cool and fluffy, then fold in the vanilla extract.
Using a butter knife, frost the bottoms of half the cookies, top them with the remaining cookies, and serve.
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