Heat oven to 400°F. In a large nonstick skillet over medium-high heat, melt 1 tablespoon butter and olive oil, then add onions. Cook, stirring until just soft, about 4 min. Add mushrooms and ½ teaspoon salt and continue stirring until softened, about 4 min. Gradually add spinach and stir until completely wilted, about 2 min. Transfer to a large bowl and add chicken; wipe skillet clean.
Whisk together flour, broth, lemon juice and ½ teaspoon salt and add to the same skillet over medium-high heat. Whisk until mixture is boiling and simmer until just thick, about 2 min. Add to chicken mixture with ½ tablespoon rosemary, then transfer to a 2-qt baking dish.
Melt 3 tablespoon butter and brush onto each phyllo sheet, layering one on top of the other. Gently place over baking dish and crimp the sides to fit. Sprinkle with remaining rosemary and bake until golden, 25 to 30 min. Let cool slightly before serving.
I used fresh spinach in this recipe, but you can swap it for a 10-oz box of frozen spinach. Thaw the spinach and squeeze out the excess liquid before using.To make this Gluten Free, swap the all-purpose flour with GF All-Purpose Flour, or 1 ½ tablespoons cornstarch. Top with Gluten Free breadcrumbs tossed in the butter, instead of phyllo.