Position an oven rack in the middle and preheat the oven to 375 degrees F. Brush the sides and bottom of a 2-quart baking dish with oil.
Combine the tofu, 2 tablespoons oil, almonds, nutritional yeast, lemon juice, garlic, 1 teaspoon salt and a few grinds of pepper in a food processor and pulse until completely smooth.
Heat the remaining 1 tablespoon oil in a large skillet over medium-high heat, then add the onions and cook, stirring frequently, until soft and deeply golden brown, 6 to 8 minutes. Remove from heat and stir in the parsley and thyme. Transfer the onion-herb mixture to a small bowl. (There's no need to wipe out the skillet; there's a lot of flavor in there.)
Combine the potatoes, almond milk, 1 cup water and 1 teaspoon salt in the skillet and bring to a simmer over medium heat. Cook, gently tossing the potatoes from the bottom to the top with a rubber spatula frequently, until the liquid has reduced and thickened and the potatoes are just slightly tender, 15 to 18 minutes.
Transfer half of the potatoes and cooking liquid to the oiled baking dish and arrange in an even layer. Season with 1/2 teaspoon salt and few grinds of pepper. Drizzle with half of the tofu mixture and sprinkle with about a third of the onion-herb mixture. Top with the remaining potatoes and cooking liquid and season with another 1/2 teaspoon of salt and a few grinds of pepper. Drizzle with the remaining tofu mixture and sprinkle with the remaining onion-herb mixture.
Cover the dish with foil and bake until just bubbly, about 30 minutes. Uncover and continue to bake until very bubbly and lightly golden brown, about 15 minutes more. Sprinkle with parsley leaves and let rest for 10 minutes before serving.
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