In glass baking dish, just large enough to hold mushrooms in a single layer, mix broth, oil, vinegar, thyme, 1/8 tsp. salt, and 1/8 tsp. pepper. Add mushrooms, turning to coat. Let stand 30 minutes, turning occasionally.
Meanwhile, from lemon, grate 1/2 tsp. peel and squeeze 1/2 tsp. juice. In small bowl, stir lemon zest, juice, soy mayonnaise, green onion, remaining 1/8 tsp. salt, and remaining 1/8 tsp. pepper.
Prepare outdoor grill or heat ridged grill pan over medium heat until hot. Add mushrooms and grill, turning occasionally and brushing with remaining marinade, until mushrooms are browned and cooked through, 8 to 10 minutes per side.
Spread four slices of toast with mayonnaise mixture; top with arugula and warm mushrooms. Cover with remaining toast.
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