To a medium bowl, add 1 cup of chopped onion, plus the bread crumbs, ¼ cup wine, 2 tablespoons olive oil, a hefty pinch of salt and a few grinds of pepper; mix until combined. Add the chicken and mix until combined. Form the meat mixture into 14 balls (about 3 tablespoons each), setting them on a sheet pan or plate.
Heat a large (12-inch) heavy skillet or braiser over medium-high, and add the remaining 3 tablespoons of olive oil.
Sear the meatballs for 2 to 3 minutes, until a dark brown crust forms on the bottoms, then flip, and continue searing for 2 minutes on the other sides. Return the meatballs to the sheet pan or plate.
Adjust heat to high. Add the remaining onions to the skillet and cook, stirring every 2 minutes, until golden, translucent and slightly darker on the edges, about 6 minutes, turning the heat down as necessary.
Add the mushrooms and cook for 6 minutes, stirring occasionally, until reduced in volume and slightly darker in color. Sprinkle the flour on top, mix to coat the mushrooms and onion and continue cooking for 1 minute, lowering the heat as needed.
Carefully pour the remaining 1 cup of wine into the skillet while constantly stirring. Using a wooden spoon, scrape the caramelized bits off the bottom of the pan and cook for 2 minutes, until the wine reaches a thicker, honey-like consistency. Add 1¾ cups of water and bring to a vigorous simmer. Season with a hefty pinch of salt.
Return the meatballs to the skillet. Adjust heat to medium, partially cover and simmer for 12 minutes, until the sauce has thickened slightly and the meatballs are cooked through.
Taste and season with salt. Sprinkle parsley on top and serve.
