Preheat the oven to 350 degrees F. Grease a 9-inch square baking dish with oil.
Heat the oil in a large nonstick skillet over medium-high heat. Add the sliced garlic, fennel seeds and red pepper flakes and stir until fragrant, about 45 seconds. Add the mushrooms and cook, undisturbed, until they begin to brown in spots, about 3 minutes. Continue to cook, stirring occasionally, until tender, about 4 minutes more. Add the tomatoes, 1 cup water and 1 teaspoon salt, bring to a simmer and cook until the sauce is reduced and most of the water is evaporated, 10 to 15 minutes. Let cool.
Meanwhile, add the tofu, spinach, vegan cream cheese, basil, nutritional yeast, whole clove garlic, 1 teaspoon salt and nutmeg to a food processor and pulse until smooth.
Spread a few tablespoons of the mushroom-tomato sauce on the bottom of the prepared baking dish and top with 4 noodles (it's OK if they overlap slightly; break 1 noodle into pieces to fill in any gaps if needed). Add half of the tofu mixture and spread out evenly. Top with a third of the sauce. Repeat with another 4 noodles, the remaining spinach mixture and half the remaining sauce. Finish the lasagna with the remaining 4 noodles and sauce. Cover with foil and bake until the lasagna is hot, about 45 minutes. Remove the foil, sprinkle with the shredded vegan cheese and bake until the lasagna is very hot and the cheese is slightly melted, about 15 minutes more. Let cool for a few minutes, cut into 6 pieces and garnish with basil leaves.
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