Heat the oil in a medium saucepan over medium heat. Add the mushrooms, garlic, green chiles and poblano. Cook stirring occasionally until the mushrooms and poblano are starting to soften. Stir in the flour and cook a minute or two. Add the spinach and continue cooking until the spinach begins to wilt, stirring occasionally. Stir in the cream and cook until starting to thicken. Season with salt and pepper.
Grease a 9” x 13” oven safe pan. Pour a bit of cream in the bottom of the pan. Arrange six tortillas to cover the pan—ok to overlap. Put a third of the spinach mixture on the tortillas. Layer half the shrimp and then grate a third of the cheese over the shrimp.Top with another layer of tortillas, spinach mixture, shrimp and cheese. Top with remaining tortillas, spinach, and cheese. Drizzle a bit of cream over all.
Cover the pan with foil and bake for ½ hour. Remove foil and bake 15 to 30 minutes more until bubbling and heated through. Let cool 15 minutes before serving (we can never wait that long.)