Make The Muhammara: Make the Muhammara Dip according to the recipe.
Bake The Chicken: Heat oven to 350°F and arrange a rack in the middle. Place chicken in a baking dish, top with 3 tablespoons of olive oil, lemon, thyme sprigs, and a big pinch of salt and a few cranks of pepper then turn to coat. Bake uncovered until chicken juices run clear and internal temperature is 165°F, about 30 minutes. Set aside until cool enough to handle then thinly slice.
Saute The Mushrooms: Meanwhile, heat 2 tablespoons of oil in a large nonstick frying pan over medium-high heat. Add mushrooms, season with salt, and cook, stirring rarely until golden brown, about 10 minutes. Season with additional salt and pepper then set aside.
Make The Chicken Melts: Heat oven to broil and arrange a rack in the middle. Spread a few spoonfuls of Muhammara Dip on one sandwich half then top with a few slices of chicken, a spoonful of the mushrooms, and a slice of cheese. Repeat to assemble 8 sandwiches then arrange on a rimmed baking sheet. Broil until cheese is melted and browned, about 3 minutes. Serve immediately.