Center a rack in the oven and preheat to 350 (F). Prepare a 12-cavity muffin tin by lining with cupcake liners.
In the bowl of a freestanding electric mixer fitted with a whisk attachment, combine 1 ounce coarsely chopped dark chocolate and 1/3 cup hot coffee. DO NOT STIR — let sit for 2 minutes to allow the chocolate to melt.
As the chocolate is melting, combine 3/4 cup all-purpose flour, 1/2 cup unsweetened cocoa powder, 1/2 teaspoon baking powder, 1/2 teaspoon baking soda, and 1/2 teaspoon kosher salt in a medium bowl and whisk until combined. Set aside.
Return to the chocolate. At this point the chocolate should be melty and easily melt into the coffee when whisked. Turn the mixer on its lowest setting and whisk the chocolate and coffee together, then add 1 cup granulated sugar, 1/2 cup crème fraîche, 1/3 cup vegetable oil, 1 large egg, and 1 tablespoon pure vanilla extract. Continue whisking until just combined.
Stop the mixer and sprinkle the dry ingredients (from the 3rd step) over the liquid ingredients (from the 4th step). Turn the mixer back on to medium-low and continue whisking until the dry ingredients are just incorporated. Use a rubber spatula to scrape down the sides and incorporate the additional batter with a one or two stirs. Use a 1-tablespoon cookie dough scoop to divide the batter evenly between the cupcake liners – see baker's notes; be careful not to overfill! I used about 2 tablespoons per cupcake. Bake in the preheated oven until the cupcakes are domed and the top springs back when gently touched, around 20 to 25 minutes. Remove from oven and allow to cool on a wire rack to room temperature before frosting.