Heat oven to 350 degrees F and arrange a rack in the lower third. Brush a 12-cup Bundt pan thoroughly with melted butter, coat with flour, and tap out the excess. Sprinkle nuts in bottom of pan and set aside.

Whisk together flour, baking powder, and salt in a medium bowl until evenly combined; set aside. Whisk together milk, ginger, and vanilla in a second small bowl until evenly combined.

Combine butter and sugar in a the bowl of a stand mixer fitted with a paddle attachment and beat on medium speed until mixture resembles wet sand and is light, about 3 minutes. Add eggs one at a time until incorporated and mixture is smooth. Scrape down mixer and sides of bowl. Reduce mixer to low, add 1/3 flour mixture and whisk until just moistened through. Whisk in 1/2 milk mixture and whisk until just moistened. Repeat until all ingredients are added, ending with flour. Transfer batter to prepared Bundt pan and smooth out so batter is even.

Bake until cake is set, golden brown, and a cake tester inserted in middle comes out clean, about 50 minutes. Transfer to a rack, let cool 10 minutes, then invert onto a wire rack, remove from pan, and let cool completely.

Combine all ingredients in a small saucepan and place over medium heat. Bring to a boil and cook until slightly syrupy, about 5 minutes. Strain and set aside to cool slightly.

When cake is at room temperature, use a skewer to poke holes all over the top. Drizzle the syrup into the holes in the cake, small spoonful by spoonful, making sure it goes into the cake and doesn't flow down. Let sit at least 4 hours so the syrup can soak into the cake.