Preheat the oven to 350°. Season the chicken generously on both sides with salt. In a large cocotte or Dutch oven over medium heat, add 3 tablespoons butter. Once bubbling, add the chicken skin side down (work in batches as needed to avoid overcrowding); cook until golden brown, about 10 minutes. Turn and repeat on the remaining side.
Remove the chicken to a baking sheet. Pour out all but 1 tablespoon of the fat from the cocotte and add the onion; season with salt and cook, turning the larger pieces occasionally and stirring the small pieces, until softened and lightly colored, about 5 minutes. Remove the large pieces to a large rimmed serving plate. Add the riesling to the cocotte, stirring to deglaze, then cook over medium-high heat until the wine is mostly evaporated, about 2 minutes. Add the stock and heavy cream. Cover and cook at a low simmer for 35 minutes.
During the last 10 minutes of simmering the sauce, place the chicken in the oven and roast until a thermometer inserted into the center of the thickest piece reads about 155°, about 10 minutes.
If frying the parsley for garnish: Set a paper-towel-lined plate next to the stove. In a small saucepan, add enough oil so that it reaches 1 inch up the sides of the pan and heat until a thermometer reads 350°. Make sure the parsley is completely dry. Add the sprigs a few at a time to the oil (stand back as it may spatter). Let cook until crisp but not brown, about 15 seconds. Using a slotted spoon, remove to the paper-towel-lined plate.
Just before serving, in a large skillet, melt 1 tablespoon butter over medium-high heat. Once bubbling, add the chanterelles and cook, stirring or shaking the pan occasionally, until browned on the edges and tender throughout, about 8 minutes. Remove to the platter with the reserved red onions. In the hot skillet, melt the remaining 1 tablespoon butter, then add the chard; season with salt and cook just until wilted, about 1 minute. Remove and distribute around the serving plate.
Taste the sauce and adjust the seasoning as needed. Strain and discard the chopped onions if desired. Pour the sauce onto the serving platter, then nestle the chicken into the sauce. Garnish with fried parsley if desired.