In a medium measuring cup with a pouring spout, combine the half and half, chicken broth, Worcestershire sauce, dried parsley, dried basil, mustard powder, dried thyme, and ground sage. Set this mixture aside. Measure the remaining ingredients before starting. When the mushrooms are almost finished cooking, begin heating the water for the ravioli.
Heat the olive oil in a deep 12-inch skillet over medium-high heat. Arrange half of the mushrooms in the skillet with enough room between them to brown rather than steam. Leave them undisturbed for 3-4 minutes on each side, adding a little more olive oil if the pan becomes too dry. Once golden brown, transfer them to a plate. Repeat the same process with the remaining mushrooms. Turn off the heat and let the skillet cool slightly.
Pour the chicken broth into the same skillet and return it to medium heat. Scrape the bottom of the skillet with a silicone spatula to release the browned bits. Allow the chicken broth to bubble gently for about 4 minutes, or until its volume has reduced by half.
Add the butter and let it melt, then stir in the minced garlic. Cook for 2 minutes. Add the flour and cook for another 1-2 minutes, stirring continuously.
Lower the heat to medium-low. Gradually pour in the prepared sauce mixture in small splashes, stirring continuously after each addition. Bring the sauce to a gentle boil, then lower the heat so it maintains a simmer.
Keep the sauce at a gentle simmer while cooking the ravioli according to the package instructions. Use a timer so the ravioli does not overcook. Drain when finished.
Turn the sauce to low heat. Stir in the Asiago and Parmesan until incorporated, then return the browned mushrooms to the skillet.
Place the cooked ravioli on a serving plate and spoon the mushroom sauce over it. Alternatively, transfer the ravioli directly to the skillet and gently stir until coated. Garnish with fresh parsley and serve.