Preheat oven to 350F. Lightly spray a 9-in. pie plate with oil.
Heat a non-stick frying pan over medium. Add oil, then turkey. Cook, stirring often to break up meat, until no pink remains, about 3 min. Add onion, garlic, mushrooms and herbs. Cook until onion is soft, about 3 min. Stir in tomato sauce and brown sugar. Gently boil until sauce reduces slightly, 5 min.
Assemble lasagna by placing a tortilla on the bottom of prepared pie plate. Spread 1 1/3 cups sauce in an even layer over tortilla. Scatter 1 1/3 cups spinach, then 2/3 cup of cheese mix overtop, pressing down gently on the tortilla as you layer. Repeat with remaining tortillas, sauce, spinach and cheese. (The spinach will cook down as it bakes.)
Bake in centre of oven, uncovered, until cheese is melted, about 20 min. Let stand for 5 min before cutting into wedges.