1. Make cake: Preheat oven to 325°F. Butter and flour a 10-inch nonstick Bundt pan.
2. In a bowl, mix flour, baking soda, ground ginger and salt. In another bowl, using an electric mixer on medium-high speed, beat butter, sugar and zest until light, about 3 minutes. Beat in eggs one at a time. Add minced ginger and vanilla.
3. Reduce mixer to low. Add one third of flour mixture, then half of buttermilk. Repeat, ending with flour mixture.
4. Transfer batter to pan. Bake until a skewer inserted in cake comes out clean, about 1 hour 10 minutes. Let cool in pan on a rack for 15 minutes. Invert onto rack to cool fully.
5. Make glaze: Whisk confectioners' sugar, lemon juice and 3 Tbsp. buttermilk until smooth. Add more buttermilk 1 tsp. at a time if needed. Drizzle over cake; let stand for 30 minutes.