GETTING READY: Center a rack in the oven and preheat the oven to 350°F Butter a 9- to 10-inch (12-cup) Bundt pan, dust the inside with flour and tap out the excess. (If you’ve got a silicone Bundt pan, there’s no need to butter or flour it.) Don’t place the pan on a baking sheet – you want the oven’s heat to circulate through the Bundt’s inner tube.
Whisk together the flour, ground walnuts, baking powder, and salt.
Set a heatproof bowl over a saucepan of gently simmering water. Put 2 tbsp of the butter, cut into 4 pieces, into the bowl, along with the chocolate, coffee and instant coffee. Heat the mixture, stirring often, until the butter and chocolate are melted and everything is smooth and creamy – keep the heat low so that the butter and chocolate don’t separate. Remove the bowl from the heat.
Working with a stand mixer, preferably with a paddle attachment or with a hand mixer in a large bowl, beat the remaining 2 sticks of butter and the sugar at medium speed for about 3 minutes – you’ll have a thick paste; this won’t be light and fluffy. Beat in the eggs one by one, beating well after each addition. The mixture should look smooth and satiny. Beat in the vanilla extract. Reduce the mixer speed to low and add the dry ingredients and the milk alternately, adding the dry mixture in 3 portions and the milk in 2 (begin and end with the dry ingredients).
Scrape a little less than half of the batter into the bowl with the melted chocolate and, using a rubber spatula, stir to blend thoroughly.
If you want to go for the gingko pattern, scrape all of the white batter into the pan and top with the chocolate. If you want a more marbled pattern, alternate spoonfuls of light and dark batter in the pan. When all the batter is in the pan, swirl a table knife sparingly through the batters to marble them.
Bake for 60 to 70 minutes, or until a thin knife inserted deep into the center of the cake comes out clean. Transfer the Bundt pan to a rack and let cool for 10 minutes before unmolding, then cool the cake completely on the rack.