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Preparation of Sauce Mixture: In a large measuring cup equipped with a spout, combine the beef broth, chicken broth, half-and-half, hot sauce, Worcestershire sauce, Dijon mustard, dried parsley, thyme, and oregano. Set aside for later use. Prepare all remaining ingredients prior to commencing the cooking process, although cheese may be grated while the onions are cooking.
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Cooking the Mushrooms: In a large skillet, heat the olive oil over medium-high heat. Add the mushrooms and sauté until they are golden brown, about 6 minutes. Season with salt and pepper to enhance their flavor, then remove from the skillet and set aside.
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Deglazing the Skillet: With the skillet off the heat, add the dry white wine. Return the skillet to medium heat and use a silicone spatula to scrape the bottom and sides of the skillet, incorporating the fond into the wine. This process adds depth to the sauce.
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Caramelizing the Onions: Add butter to the skillet along with the sliced onions. Cook over medium heat, stirring frequently, for 10 minutes. Then reduce the heat to medium-low and continue cooking, still stirring frequently, until the onions are fully caramelized, about 35 additional minutes. Adjust the heat as necessary to prevent the onions from burning and add a little more olive oil or wine if more moisture is needed.
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Preparing the Pasta: While the onions are caramelizing, bring a large pot of water to a boil. Add a generous amount of salt to the water (approximately ½ tablespoon of kosher salt is recommended).
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Incorporating Garlic and Flour: Add the minced garlic to the caramelized onions and cook for 1 minute. Sprinkle the flour over the onions and garlic, cooking for an additional 2 minutes while stirring continuously to form a roux, which will help thicken the sauce.
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Adding the Sauce Mixture: Gradually add the prepared sauce mixture to the skillet in small increments, stirring continuously. Bring the mixture to a gentle boil, then reduce the heat and allow it to simmer, partially covered, while the pasta cooks.
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Cooking the Pasta and Final Assembly: Cook the orecchiette pasta in the boiling water until al dente, according to the package instructions. Drain the pasta and then reduce the heat under the sauce to low. Gradually add the shredded Gruyere and grated Parmesan cheeses to the sauce, stirring continuously until the cheeses are fully melted and incorporated. Add the sautéed mushrooms back into the sauce. Gradually incorporate the cooked pasta into the sauce, adjusting the sauce-to-pasta ratio according to preference. Serve immediately, garnished with additional Parmesan cheese if desired.
