Rinse chicken and pat dry. With a meat mallet or the side of a heavy cleaver, pound chicken to the thickness of two stacked nickels.
Combine the flour, salt, pepper, and dill. Sprinkle (do not dredge) the chicken with the mixture.
Heat the olive oil until it shimmers in a skillet over medium-high heat. Sear the chicken until it lightly browns on both sides. Remove and keep warm.
Add 1/4 cup of water to the pan and bring to a boil. Whisk the bottom of the pan to dissolve the olive oil and little bits of browned chicken that may have stuck. Lower the heat to medium-low. Add the spinach, still dripping its wash water, and the garlic. Cook until the spinach softens, stirring carefully (avoid breaking the leaves). After cooking, drain the spinach well.
Coat the inside of a glass baking dish or casserole (about 9 x 5 x 2 inches) with olive oil. Place the thinnest chicken slices on the bottom. Top with slices of the cheese, then the spinach. Sprinkle a little of the lemon juice over this, then create another layer in the same way. Top that with the remaining lemon juice, and then with the phyllo pastry.
Put the baking dish into the 350-degree oven for 25 minutes, or until a knife poked into the center shows that the inside is hot. The pastry should also be browned by this time.
Allow to cool for five minutes. With a sharp knife, cut into squares or rectangles. The best utensil for extracting the first cube is a cake-frosting knife. Serve with two tablespoons of tzatziki sauce.