Preheat oven to 400°F. Lightly coat a 9-by-13-inch baking dish with cooking spray.
Heat 1 tablespoon oil in a large skillet over medium heat. Add onion; cook, stirring, until softened, about 5 minutes.
Add garlic; cook until fragrant, 1 minute.
Stir in spinach and flour; cook, stirring, until combined and mostly dry, about 2 minutes.
Add artichokes, broth and cream cheese; stir until thickened, 1 to 2 minutes.
Remove from heat. Stir in chicken, lemon zest, lemon juice and feta.
Pour the mixture into the prepared baking dish.
Place 1 phyllo sheet on top of the chicken mixture; brush with some of the remaining 2 tablespoons oil.
Repeat the process with the remaining phyllo sheets and oil, stacking one on top of the other.
Score the top of the casserole with a knife to make 8 slices.
Bake until the top is golden brown, about 30 to 35 minutes.
Let cool for 10 minutes before serving.
