Make the cupcakes:
Preheat the oven to 350°F Line a standard 12-cup muffin tin with cupcake liners.
To make the cupcakes, in a very large saucepan, bring the stout, molasses, and oil to a boil over medium-high heat. Remove from the heat and whisk in the baking soda until dissolved. (The mixture will foam up, then settle down.) Stir in the brown sugar, then let cool until tepid.
Into a small bowl, sift together the flour, baking powder, ginger, cinnamon, cloves, and salt.
Whisk the eggs into the stout mixture, then whisk in the flour mixture just until incorporated. Don’t overmix. Gently stir in the minced candied ginger.
Divide the batter among the cupcake liners and bake until the cupcakes feel just set in the center, 22 to 24 minutes. Let cool completely.
Make the frosting:
In a stand mixer fitted with the paddle attachment, beat the butter on high speed until smooth, about 10 seconds. Decrease the speed to low and, with the mixer running, gradually add half of the confectioners' sugar. Stop the mixer and scrape down the bowl as needed to make sure the ingredients are being incorporated. Add the lime juice, then add the remaining confectioners' sugar. Once the sugar is incorporated, add the milk. Beat the frosting on high speed until completely smooth and fluffy, about 3 minutes. Taste, and add a few more drops of lime juice, if desired.
Transfer the frosting to a pastry bag fitted with a star tip. Remove the cupcakes from the muffin tin. Pipe rosettes of frosting in the center of each cupcake. (If you don’t have a pastry bag, you can spoon a mound of frosting decoratively in the center.) Garnish each with strips of candied citrus peel or a piece of candied ginger.