This is Taiwan's national dish. The "three cups" refer to the primary ingredients: soy sauce, rice wine and sesame oil. I must admit, I use less than a full cup of sesame oil! It is originally a Chinese dish, but very popular in Taiwan. This version, with Thai basil, is made in the Taiwan style. The traditional Jiangxi recipe has no basil.
Cut the wings into three sections. Save the wingtips for stock or discard them. Pat the drum and flat sections dry with paper towels.
Rehydrate the mushrooms in warm tap water, about 15 minutes. When softened, remove the stems and cut the caps into quarter-moon pieces. Discard the stems.
In a heavy-bottomed pot over medium-high heat, add the oil, ginger slices, garlic, chiles and white sections of the green onions. Stir fry until aromatic and beginning to soften, about 2 minutes.
Move the aromatics to one side of the pan and add the chicken pieces in a single layer, working in batches if necessary. Cook each piece of chicken until golden brown on all sides, turning occasionally. If the aromatics start to get too dark, remove them to a small plate and reserve.
When the chicken is golden, add the rice wine, thin soy sauce, sweet soy sauce, sesame oil, sugar and mushroom caps. Add the aromatics back to the pot if you removed them. Bring to a boil and reduce to a simmer. Loosely cover and simmer for 30 minutes, stirring occasionally so all the pieces get color and flavor.
If the cooking liquid is thin, remove the lid, raise the heat to medium and allow the sauce to reduce. Stir frequently to prevent burning. The bubbles will get larger and start to glaze onto the chicken pieces. When the sauce has reduced to a rich glaze, taste and adjust the seasonings if necessary. Garnish with the green onion tops and Thai basil if using. Serve immediately.