In a medium skillet over medium heat, melt butter and olive oil.  Add smashed garlic cloves and the whole sprigs of rosemary.  Sauté until fragrant, about 2 to 3 minutes.  Add mushrooms, salt, and pepper.  Sauté about 10 to 12 minutes, stirring often, until all of the liquid has been absorbed and the mushrooms are tender.  Remove garlic cloves and rosemary.  Set aside.

Spoon the mushroom mixture onto one of the rectangles, leaving a 1-inch border all around.  Using a pastry brush, coat the 1-inch border of pastry with beaten egg.  Lay the remaining rectangle of pastry over the filling and press gently to seal.  Leave the pastry on the parchment-lined baking sheet and refrigerate another 20 minutes (or up to 2 hours).

Preheat oven to 375 degrees F.

Using a small knife, trim the edges of the pastry. Using a pastry brush, coat the entire top of the pastry with the beaten egg.  Gently cut five 3-inch vents in the top, being careful not to pierce the bottom layer.  Bake, rotating once, for about 35 to 40 minutes, until the crust is deep golden brown and the pastry has puffed.  Using one or two spatulas, transfer the pie to a wire rack and let cool for about 10 minutes before slicing and serving.  The pie also tastes great at room temperature.

In a food processor, combine the garlic, serrano, avocados, lime juice, salt, and pepper. Pulse until chunky. Transfer to a bowl and stir in cilantro. Serve with tortilla chips.

Heat the canola oil over medium-low heat in a wok or large skillet. Add the garlic and red pepper flakes. Let garlic and red pepper flakes infuse the oil for 3 to 4 minutes. Increase the heat to medium and add the broccoli. Stir-fry for 3 to 4 minutes. Add the stock, soy sauce, and black pepper, and cook uncovered for about 5 minutes, until broccoli is tender and almost all the liquid has evaporated.

Season the chicken with 1 tablespoon salt and 1 teaspoon pepper. Place 1 cup flour in a large dish. Dredge all the chicken thighs until coated in flour. Set aside in a single layer.

Heat a large Dutch oven over medium-high heat. Add the oil. Sear the chicken in batches, skin side down, for 4 to 5 minutes, until evenly browned. Do not turn the chicken or move it while it is browning, to encourage a good, golden crust on the skin. Remove the chicken to a plate and reserve.

Reduce the heat to medium and melt butter. Add the garlic and mushrooms. Cook until the garlic is golden brown and any liquid from the mushrooms has evaporated, 7 to 8 minutes. Add the rosemary and thyme and cook an additional 2 minutes. Stir in the white wine and chicken stock. Return the chicken to the Dutch oven. It is fine if the chicken is overlapping or in two layers. Bring the liquid to a low boil, then reduce to a gentle simmer. Cover and let cook 20 minutes. Remove the chicken to a platter and tent with foil to keep warm. Stir in cream, lemon juice, and remaining salt and pepper into the juices in the pot. Whisk the remaining 3 tablespoons flour with the water and stir into broth. Return the liquid to a low boil and cook until thickened, about 5 minutes. Serve the chicken with the gravy.