In a skillet, melt butter over medium heat. Add shallots and sauté until tender and translucent but not browned, about 6 minutes. Stir in garlic and asparagus and cook 2 minutes. Transfer to bowl and let cool. Mix in goat cheese, parsley, lemon zest, salt and pepper.
With knife held horizontally, slice through chicken breasts, starting at the thinner side, about two-thirds of the way through, so chicken breasts can open like a book. Spread one-quarter of the filling over the inside of each breast leaving a 1/2-inch border uncovered. Close each breast to enclose filling, securing with a toothpick if desired.
Heat oven to 400°F
In a large ovenproof nonstick skillet, heat oil over medium-high heat. Sprinkle the chicken breasts with half of the salt and pepper. Arrange chicken, skin-side down, and cook until golden brown, about 3 minutes. Gently turn chicken over and cook 3 minutes more.
Transfer skillet to oven and bake until chicken is cooked through, about 15 to 20 minutes. Transfer chicken to a platter to keep warm.
In same skillet, melt butter over mediumhigh heat. Add shallots and cook until tender but not browned, about 6 minutes. Stir in wine and chicken broth. Bring to a boil. Reduce heat to maintain a simmer for 3 minutes. Whisk yolks and remaining salt and pepper in a small bowl. Pour about 1/4 cup of the sauce into the yolks and whisk to combine. Whisk yolk mixture back into skillet and continue whisking, without letting it boil, until sauce is thick enough to coat the back of a spoon, about 3 minutes. Stir in parsley and tarragon; serve over chicken.