How to Make It
In a large saucepan, combine all of the ingredients with a generous pinch each of salt and pepper. Cover with water and bring to a boil over high heat. Simmer over moderate heat until the rice is tender, about 1 hour. Drain well.
Set a rack over a large rimmed baking sheet. In a pot, heat the oil. Add the shallot and garlic and cook over moderately high heat, stirring, until softened, 1 to 2 minutes. Add the Swiss chard in large handfuls, letting each batch wilt slightly before adding more. Cook, stirring occasionally, until all of the chard is wilted, 8 to 10 minutes. Spread the chard out on the rack to drain and let cool completely. Squeeze out any excess water.
In a large, deep skillet, heat 2 tablespoons of the oil until shimmering. Add half of the mushrooms and cook over moderately high heat, undisturbed, until browned on the bottom, 5 minutes. Cook, stirring, until the mushrooms are tender and browned all over, 5 minutes longer; transfer to a plate. Repeat with the remaining oil and mushrooms.
Wipe out the skillet and melt the 2 tablespoons of butter in it. Add the onion, celery, garlic, thyme, rosemary and a generous pinch each of salt and pepper. Cook over moderate heat, stirring occasionally, until the vegetables are just starting to brown, about 8 minutes. Stir in the mushrooms. Sprinkle the flour over the vegetables and cook, stirring, until incorporated, about 2 minutes. Gradually whisk in the stock and bring to a boil, stirring frequently. Reduce the heat to moderate and simmer, stirring occasionally, until the sauce is thickened and no floury taste remains, about 7 minutes. Stir in the cream and season the sauce with salt and pepper.
Preheat the oven to 375°. Arrange half of the chicken in the bottom of a 9-by-13-inch or 4-quart baking dish that’s at least 2 inches deep. Scatter half of the Swiss chard over the chicken, followed by half of the wild rice and half of the mushroom cream sauce. Repeat the layering once more with the remaining chicken, greens, rice and sauce.
In a medium bowl, toss the panko with the 3 tablespoons of melted butter and sprinkle evenly over the casserole. Cover with foil and bake for about 35 minutes, until bubbling. Uncover the casserole and turn on the broiler. Broil 6 inches from the heat until the panko is lightly browned, about 3 minutes. Let stand for 10 minutes. Garnish with chopped parsley and serve.