1. Preheat the oven to 425°. Spread the chicken wings in a roasting pan and season with salt. Bake for about 45 minutes, or until deeply browned.
2. Meanwhile, in a heatproof bowl, cover the dried porcini with the boiling water and let soak until softened, about 20 minutes. Rub the porcini to loosen any grit, then lift them out and transfer to a plate. Let the soaking liquid stand for 5 minutes so the grit settles.
3. In a large saucepan, heat the oil until shimmering. Add the onion and cook over moderately high heat, stirring occasionally, until browned, about 5 minutes. Add the mushroom stems and cook until browned, about 5 minutes. Add the porcini and their soaking liquid, leaving behind any grit. Stir in 3 quarts of water and bring to a boil.
4. Transfer the chicken wings to the saucepan. Set the roasting pan over 2 burners and cook over moderate heat until sizzling. Add 1 cup of water and bring to a simmer, scraping up the browned bits. Pour the pan juices into the saucepan and simmer over low heat, skimming occasionally, until reduced to 5 cups, about 2 hours. Strain the broth and skim off the fat.
5. In a large skillet, melt the butter over high heat. Add the oyster mushroom caps and season with salt and pepper. Reduce the heat to moderate, cover and cook, stirring occasionally, until softened and lightly browned, about 7 minutes. Uncover and cook, stirring, until well browned, about 4 minutes.
6. Bring the mushroom broth to a simmer and season with salt and pepper. Run a thin knife around the edge of each Parsnip Custard. Gently invert the custards, 1 at a time, onto a large spatula and transfer each to the center of a shallow bowl. Or, cut the large custard into 10 squares and place in the bowls. Ladle the broth around the custards, garnish with the mushrooms and serve.
Make Ahead: The broth and mushrooms can be refrigerated separately overnight. Reheat before serving.