Add the butter to a 6-quart Instant Pot® and set to high saute (see Cook's Note). When melted, add the mushrooms and 2 teaspoons salt and cook, stirring occasionally, until golden brown, 8 to 10 minutes. Stir in the flour and nutmeg and cook for 1 minute. Whisk in the milk, bring to a boil and cook until thick and creamy, about 5 minutes. Transfer the bechamel sauce to a medium bowl and set aside. Add 2 cups water to the pot (no need to clean the pot first) and put the rack in the bottom with the handles folded under.
Meanwhile, mix together the spinach, basil, ricotta, red pepper flakes, garlic, 1 cup Parmesan and 1 teaspoon salt in a medium bowl until completely combined.
Spread 1/2 cup marinara sauce on the bottom of an 8-inch round cake pan. Break 4 lasagna noodles into large pieces and arrange in an even layer over the sauce. Spread another 1/2 cup marinara sauce evenly over the noodles. Layer a third of the bechamel over the marinara and top with a third of the spinach mixture. Top with another layer of 4 broken noodles. Repeat the process 2 more times, starting with 1/2 cup marinara and ending with broken noodles, for a total of 4 layers of noodles. Top with the remaining marinara and cover tightly with foil, wrapping any excess underneath (if you bunch the foil on the side of the pan it won't fit in the pot).
Fold an 18-inch piece of foil in half lengthwise, then in half again lengthwise and finally once more in half lengthwise, making a 4 1/2-inch-wide foil sling for the pan. Place the cake pan with the lasagna in the center of the sling and lower into the pot.
Follow the manufacturer's guide for locking the lid and preparing to cook. Set to pressure cook on high for 25 minutes.
After the pressure-cook cycle is complete, follow the manufacturer's guide for natural release. After 15 minutes, being careful of any remaining steam, unlock and remove the lid.
Position a rack in the upper third of the oven and preheat the broiler to high.
Remove the foil from the lasagna and top with the mozzarella and remaining 1/4 cup Parmesan. Broil until the cheese is melted, bubbling and brown in spots, 1 to 4 minutes. Let sit 10 minutes before serving.
Settings may vary on your Instant Pot® depending on the model. Please refer to the manufacturer's guide.
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