Preheat oven to 425°F. Dry chicken well and season generously with salt and pepper. Heat a 5-quart braiser or other large, heavy, oven-proof skillet over medium heat. Melt 1 tablespoon butter in the pan and add chicken in a single layer. Cook, turning once, until the edges are lightly browned, about 4 minutes. Chicken will not be fully cooked at this point. Remove from heat and set aside.

Melt 2 tablespoons butter in same skillet, add onions and season lightly with salt and pepper. Cook, covered, for 10 minutes, stirring occasionally, until onions soften. Then uncover and continue cooking, stirring often, until onions turn light golden brown, about 10 to 15 minutes longer. Add a splash of water every so often if bottom of pan gets too dark. Add garlic, thyme, wine, and chard. Cook until garlic is fragrant and chard is wilted, about 3 minutes. Sprinkle in flour and cook, stirring, for 1 minute longer, then stir in chicken stock and milk. Bring to a simmer and cook until mixture is thick enough to coat the back of a spoon. Turn off heat, taste it for seasoning and add salt and pepper, to taste. Return chicken to skillet, add scallions, and mix well. Dot top with crumbled goat cheese.

Melt the remaining 3 tablespoons butter in a small saucepan or the microwave. Place phyllo sheets on counter under a lightly damp kitchen towel. Lay one sheet of phyllo on work surface and brush with melted butter, then scrunch up the sheet and place it on top of the mixture in the skillet, the more folds the better. Repeat with remaining phyllo until the skillet is completely covered. Drizzle any remaining butter over the top of the dough. Using a sharp knife, poke several holes straight down through the dough.

Bake until phyllo is golden and crisp on top, about 20 to 25 minutes.