To ensure that the lasagna doesn't come out watery, it's important that the greens go in dry. Let all the water evaporate during cooking, and then blot the leaves with a paper towel if necessary after removing them from the saute pan.
Heat a large nonstick skillet over high heat. Add 1 teaspoon oil and half the mushrooms. Saute, stirring occasionally, until mushrooms are golden brown and tender, about 8 minutes. Remove from heat, and let cool. Repeat with 1 more teaspoon oil and the remaining mushrooms.
Chop 6 garlic cloves. Heat 1 teaspoon oil over medium heat in a large saute pan or skillet. Add garlic and 1/2 teaspoon salt, and cook until fragrant, about 1 minute (do not let brown). Raise heat to medium-high, and add 1/2 cup water. Add two-thirds of the kale; cover, and cook until wilted, about 1 minute. Add remaining kale. Cover, and cook 2 minutes. Stir in escarole, and add a little more water if pan is dry. Cover, and cook until greens are wilted and tender, about 4 minutes. Remove lid, and cook, stirring frequently, until liquid evaporates.
Preheat oven to 375 degrees. Coat bottom of a 9-by-13-inch ovenproof baking dish with 1 teaspoon oil. Place beans in a food processor. Finely grate remaining garlic clove over beans. Puree 1 minute. Scrape down sides of bowl, add reserved bean liquid, and puree until smooth, about 1 minute. Add oregano, and pulse until just combined.
Coarsely chop tomatoes, reserving juices. Combine tomatoes and juices with red-pepper flakes, and spread one third in baking dish. Top with half the greens. Spread half the bean puree over the greens with a small offset spatula, and sprinkle with a third of the cheese. Scatter evenly with three-quarters of the mushrooms.
Continue layering with a third of the tomatoes, the remaining greens, and a third of the cheese. Dot top with spoonfuls of remaining bean puree, using the back of the spoon to slightly flatten mounds (it will spread as it bakes). Layer with remaining mushrooms, tomatoes, and cheese. Cover with foil, being careful not to let foil touch surface, and bake until bubbling around edges and hot but not bubbling in center, 30 to 35 minutes.
Turn on broiler. Transfer lasagna to broiler, and cook until top just turns golden brown, 1 to 2 minutes. Let cool 5 to 10 minutes before serving.
No Meat Required Cremini mushrooms add satisfying heartiness to this vegetarian lasagna without packing in calories or fat.