Heat oven to 400°F Cook the bulgur according to the package directions; transfer to a medium bowl and let cool to room temperature. Stir in the almonds, scallion whites, thyme, 1/2 tsp salt, and 1/4 tsp pepper.
Meanwhile, place the mushrooms on a rimmed baking sheet, rub with 2 Tb of the oil, and season with 1/4 tsp each salt and pepper. Roast, stem side up, until tender, 18-20 minutes. Dividing evenly, top the mushrooms with the bulgur mixture and sprinkle with the Feta. Roast until warmed through, 2 to 4 minutes more.
In a large bowl, mix together the lemon juice, the remaining 2 Tb of oil and 1/4 tsp each salt and pepper. Add the greens and cucumber and toss to combine. Drizzle the mushrooms with olive oil, sprinkle with the scallion greens, and serve with the salad.