Measure the mustard into a small bowl or glass measuring cup. Whisk in the vinegar. Gradually whisk in 5 tablespoons olive oil. Mix in the minced shallot and thyme. Season to taste with salt and pepper.
Coat a small baking sheet with olive oil spray. Place the fish skin side down on the sheet. Sprinkle with salt and pepper. Transfer 2 tablespoons of the vinaigrette to a small bowl and brush on the fish. Let the fish marinate while preparing the salad.
Preheat the oven to 425°F. Cook the green beans in a pot of boiling salted water until crisp-tender, about 5 minutes. Drain. Soak in a large bowl of cold water to stop the cooking, and then drain well. Pat dry. Cut the frisee into 2-inch lengths. Combine 2 cups of the frisee and the green beans in a salad bowl.
Heat 1 tablespoon of olive oil in a large nonstick skillet over medium heat. Add the sliced shallot and sauté 1 minute. Add the mushrooms, sprinkle with salt and pepper and sauté until just tender, about 4 minutes. Remove from the heat.
Meanwhile, place the fish in the oven and roast until just springy to the touch and appears opaque when cut into with a small, sharp knife, about 8 minutes.
Add 2 tablespoons of the vinaigrette to the green beans and frisee and toss to coat. Add the mushrooms and toss gently. Transfer the fish fillets to 2 plates. Mound the salad alongside and serve.