Preheat the oven to 425 degrees . Slice the mushrooms 1/4 inch thick. In a medium skillet, melt 1 1/2 tsp. butter over medium heat. Add half of the mushrooms and cook until golden, about 5 minutes. Remove and let cool. Repeat with 1 1/2 tsp. butter and the remaining mushrooms.
Melt 3 tbsp. butter over medium heat; add the onion and cook, stirring, for 3 minutes. Whisk in the flour and cook for 2 minutes. Whisk in 1 cup chicken broth; cook until thickened. Add the remaining 1 cup broth; boil. Reduce the heat, stir in the mushrooms and peas and simmer, whisking until thickened, 2 minutes. Season and remove from the heat; cover.
In a shallow bowl, whisk together the eggs, a 1/2 cups water and 1/4 tsp. pepper. In a microwave, melt the remaining 4 tbsp. butter, about 30 seconds. Lay the bread slices flat and cut a horizontal pocket into the center of each without cutting all the way through. Brush both sides of the bread with melted butter, then dip in the eggs, turning to soak. Transfer to a parchment-lined baking sheet and stuff with some of the chicken. Bake until golden-brown, 25 to 30 minutes. Top with the mushroom sauce.