Fill a saucepan with enough salted water to cover the potatoes and bring to a boil. Reduce the heat to medium and continue to cook uncovered until they are soft, about 30 minutes.
Drain the potatoes from the pot and pass them through a ricer, onto a clean cutting board. Prepare a large stockpot with about 6 quarts of salted water and bring to a boil. Fill a bowl with ice water and keep in close proximity to the stockpot.
Form a well in the middle of the potatoes and scatter with the flour. Add the egg to the middle of the well and mix with a fork. As you are mixing, add the cheese, chives, basil, cilantro, salt and pepper. Work the mixture until it reaches a smooth consistency. Knead the dough until it is dry enough so that it does not stick to the cutting board. Be careful not to overwork the dough.
Place the dough into a pastry bag with a 1/2-inch tip and press out bite-sized pieces. Place the pieces on a tray lined with parchment paper, making sure that they do not stick together. Return to the stockpot and in small batches, drop the gnocchi into the boiling water.
When the gnocchi float to the top, transfer them to the bowl of ice water to stop the cooking. Remove the gnocchi from the ice water and place them into a clean pan. Drizzle with the olive oil to prevent the gnocchi from drying out.
When ready, heat a large saute pan over high heat. Add the oil and swirl to coat the bottom of the pan. When the oil shimmers, add about 20 of the gnocchi and quickly sear, about 2 minutes on each side. Remove the gnocchi using a spatula and allow them to drain on a paper towel.
Plating: Portion out the gnocchi and place them in the center of the plate. Place the mushroom ragout on top of the gnocchi and sprinkle with the tomatoes and the lemon zest. Drizzle the Asian pesto around the outside of the plate.
Wild Mushroom Ragout:
Heat a heavy medium saucepan over medium heat. Add 1 tablespoon of the butter and swirl to coat the bottom of the pan. Add the garlic and shallots and saute, stirring until soft, about 5 minutes. Add all the mushrooms and saute until soft, 8 to 10 minutes. Season with salt and pepper, to taste. Add the wine, stir, and cook until the wine is reduced by half, about 5 minutes. Add the stock, demi-glace, dark soy sauce, and thyme, and simmer until reduced by half, about 25 to 30 minutes. Stir in the remaining 2 tablespoons of butter and correct the seasonings. Set aside.
Asian Hazelnut Pesto:
In a food processor, combine the hazelnuts, garlic, cilantro, mint, Thai basil, olive oil lemon juice, salt and pepper. Puree for about 2 minutes or until all the ingredients are incorporated.
Leftovers: Pesto Soup with Mushroom Dumplings:
In a saucepan, heat the stock and whisk in the pesto. Check for seasoning. Meanwhile, lay out 12 wonton skins and place a small mound of the ragout in the center. Top with basil or cilantro leaf. Egg wash the perimeter of each wonton and top with second skin. Press down hard on all sides to seal. In a large pot of boiling, salted water, blanch the raviolis for only 3 to 4 minutes until the skins become translucent.
Plating: In 4 large soup plates, place a small mound of spinach with the tomatoes and season with salt, pepper, and lemon juice. Place 3 raviolis on top and ladle in the hot broth. Serve with crusty bread.