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.