How to Make It
Stir together olive oil, 1 tablespoon salt, paprika, thyme, 1 teaspoon black pepper, and cayenne in a small bowl. Rub mixture on outside and under skin of chicken. Stuff cavity with lemon, garlic cloves, and 4 onion quarters.
Place 2 inverted shallow ramekins in a 6-quart slow cooker. Place carrots, celery, and remaining onion quarters around ramekins; pour water over vegetables. Place chicken, breast side up, on top of ramekins. Cover and cook on HIGH until a thermometer inserted into thickest portion of breast registers 165°F, about 3 1/2 hours. Transfer chicken to a cutting board, and let rest 15 minutes.
Preheat broiler to HIGH with oven rack 6 inches from heat. Remove ramekins, and pour stock mixture through a wire-mesh strainer into a bowl; discard solids. Reserve stock for later use. Cut chicken into 6 pieces, and place, skin side up, on an aluminum foil-lined rimmed baking sheet. Broil until skin is golden brown and crispy, 4 to 6 minutes. Set chicken aside.
Heat 2 tablespoons unsalted butter in a 10-inch cast-iron skillet over medium and cook potatoes, chopped leeks, and chopped carrots until slightly softened, about 8 minutes. Add mushrooms, 2 teaspoons kosher salt, chopped garlic, and 3/4 teaspoon black pepper. Cook until potatoes and mushrooms are tender, about 8 minutes, adding water, 1 tablespoon at a time, if vegetables stick to skillet. Stir in 12 ounces shredded, skinless Chicken and thawed green peas; remove from heat.
Melt 6 tablespoons butter in a saucepan over medium heat and whisk in flour; cook, whisking constantly, 1 minute. Gradually whisk in chicken broth (or reserved strained stock); cook until thickened, about 6 minutes. Add heavy cream and hot sauce. Stir cream mixture into chicken mixture. Place the thawed puff pastry sheet over filling, letting corners drape over edge. Brush lightly with egg wash (from 1 large egg), and cut 3 to 4 slits in top of pastry sheet. Bake in preheated oven until golden brown and bubbly, 20 to 22 minutes. Let stand 15 minutes.