Preheat oven to 375˚F. Grease a muffin tin or line with paper liners. Set aside.
Slice the ginger into large chunks, leaving the skin on. If you have a food processor, use it to finely mince into tiny pieces. If not, mince by hand. Measure out 1/4 cup, gently packing it to measure. Combine the ginger and 1/4 cup of the sugar in a small skillet and cook, stirring, over medium heat, until the sugar has melted and the mixture is hot, just a few minutes. Remove from heat and set aside.
In a small bowl, combine the lemon zest and 3 tablespoons of the sugar. Add it to the ginger mixture in the skillet.
Beat the butter in a mixing bowl or with a stand mixer. Pour in the last 1/2 cup of sugar, and beat until smooth and incorporated. Add the eggs, one at a time, beating well after each one. Pour in the buttermilk and beat until blended. Add the flour, salt, and baking soda, and beat until smooth. Beat in the ginger and lemon zest mixture, until well-incorporated.
Scoop the batter into the muffin tin, and bake for 18-20 minutes, until a toothpick comes out clean. Delicious served warm!