Chop eggplant and, in large bowl, toss with 1/2 tsp salt.

In large skillet (or skillets) over medium, heat 1 tsp olive oil. Add half the mushrooms. Sprinkle with 1/8 tsp salt, 1/8 tsp pepper, and 1/2 tsp thyme. Cook until liquid has almost evaporated and mushrooms are browned on both sides, about 4 minutes. Transfer to plate and repeat with remaining mushrooms. Add more oil only if needed.

In large pot, heat remaining 2 tsp olive oil over medium heat. Add onions and green onion whites (reserve greens for later) and cook, stirring occasionally, until caramelized, about 17 minutes. Reduce heat if onions are turning black at edges, adding 1 Tbsp water if too dry.

Preheat oven to 350 F

Rinse eggplant quickly, then, using paper towel, gently squeeze eggplant to remove juices. Add eggplant to caramelized onions along with gingerroot and bay leaves. Stir for 1 minute. Add vegetable stock and half of arame and simmer over medium-high heat, stirring frequently, about 8 minutes, until eggplant is tender and fairly dry. Stir in 4 1/2 tsp lime juice and 3 tsp Dijon. Cook for 1 minute. Add more Dijon or up to 1 1/2 tsp more lime juice—the mixture should have a sharp flavor. Remove from heat and use stick blender to blend to rough purée.

Grease 8 inch springform pan or 8 inch round or square casserole dish. Press eggplant mixture into bottom and up sides. Bake for 25 minutes.

With 10 minutes remaining, in large bowl, beat egg whites with cream of tartar to stiff peaks. In small bowl, whisk egg yolks with remaining 1/2 cup arame, 1 tsp Dijon, 1 1/2 tsp fresh thyme, reserved green onion greens, 1 1/2 tsp lime juice, plant-based milk, parsley, and 1/2 tsp salt. Fold 1/2 cup beaten egg whites into egg yolk mixture, then fold into remaining egg whites. Pour on top of eggplant crust. Gently lay mushrooms on top of egg mixture.

Bake for 35 to 40 minutes, until eggs are set.