In large saucepan of boiling salted water, cook noodles for 2 minutes less than package instructions for al dente; drain. Arrange noodles, keeping edges from touching, in single layer on tea towel.
While noodles are cooking, in nonstick skillet, heat oil over medium-high heat; cook mushrooms, stirring occasionally, until softened and almost no liquid remains, about 7 minutes. Add spinach; cook, stirring, until beginning to wilt, about 2 minutes. Drain in colander; scrape into bowl. Stir in salt and pepper.
In small saucepan, whisk 1/4 cup of the milk with the our until smooth. Gradually whisk in remaining milk; cook over medium heat, whisking constantly, until thickened, about 8 minutes. Add 3/4 cup of the mozzarella, the ricotta, mustard and cayenne pepper; cook, stirring, until mozzarella is melted. Remove from the heat.
Spoon 1/2 cup of the sauce into bottom of lightly greased 10-inch cast-iron or ovenproof skillet. Arrange 3 of the noodles over top, trimming ends to fit skillet; top with half of the mushroom mixture. Spoon half of the remaining sauce over top. Arrange remaining noodles, perpendicular to bottom noodles, over top. Top with remaining mushroom mixture, sauce and mozzarella.
Bake in 425°F oven until mozzarella is melted, about 4 minutes; broil until top is golden, about 2 minutes.