Put the mushrooms in a large baking dish. Add the shallots, lemon juice, thyme, salt, some pepper and the Tbsp of oil; toss the mixture to coat the mushrooms in a single layer, then roast them until they are tender and most of the liquid has evaporated, about 20 to 25 minutes.
Remove the mushrooms from the oven and keep dish warm.
While the mushrooms are cooking, make the dressing. Combine the mustard, vinegar, parsley, oregano and some pepper in a small bowl. Whisking vigorously, pour in the Tbsp of oil in a thin, steady stream. Continue whisking until the dressing is well combined; set the dressing aside.
Pour enough water into a saucepan to fill it about 1 inch deep. Set a vegetable steamer in the pan and bring to a boil. Put the broccoli florets into the steamer, cover the pan, and steam the broccoli until it is tender but still crisp - about 4 minutes.
Add the broccoli to the dish with the mushrooms. Pour the dressing over the vegetables and toss the salad well.
Arrange the salad on a bed of the lettuce leaves; it may be served warm or chilled