Pat the duck breasts dry with paper towels. Season both sides evenly with kosher salt.
Heat 1 tablespoon of avocado oil in a large 12-inch cast-iron skillet over medium-high heat until shimmering. Add 1 tablespoon of butter and the garlic, then sauté for 30 seconds until fragrant.
Stir in mushrooms, ½ teaspoon salt, and ¼ teaspoon black pepper and cook for 4 minutes. Add rosemary and thyme and cook for another 1 minute. Use a spatula or spoon to transfer the mushroom and garlic mixture to a plate and set aside. Discard the sprig of rosemary.
Return the skillet to high heat for 2 minutes. Add the remaining 2 tablespoons of avocado oil. Once the skillet is hot and the oil is shimmering, carefully place the duck breasts, skin-side down, in the skillet, then immediately reduce the heat to medium. Using tongs, gently press each breast against the skillet for the first minute so the skin makes full contact with the pan. Continue cooking until the skin is deeply golden, crisp, and releases easily from the skillet, about 3 minutes longer. Carefully turn the duck breasts over and sprinkle the crispy skin with a pinch of kosher salt. Continue cooking until the thickest part of each duck breast reaches 125 degrees F on an instant-read thermometer, about 2 minutes longer. Transfer the duck breasts to a cutting board and let them rest, skin side up, for about 5 minutes while you cook the asparagus. Resting allows the juices to redistribute throughout the meat. Keeping the duck skin side up helps preserve its crisp texture until it's time to serve.
Add the asparagus, ¼ teaspoon salt, and a few grinds of black pepper to the skillet. Sauté, stirring often, for 3 to 5 minutes, or until tender. The exact timing will depend on the thickness of your asparagus. Push the asparagus to one side of the skillet, then return the mushrooms along with the remaining 2 tablespoons of butter. Cook for another minute, just until the butter melts and the mushrooms are heated through again.
Divide the asparagus and mushrooms evenly among four plates, making sure everyone gets a few of those buttery garlic cloves.
Serve the duck breasts whole, or slice them thinly across the grain on a slight diagonal with a sharp knife, and arrange them over the asparagus. Enjoy immediately while the skin is still wonderfully crisp.