How to Make It
In a medium saucepan, toast the barley over high heat, stirring a few times, until browned and fragrant, about 5 minutes. Add the vegetable stock and oil, cover and simmer over moderately low heat for 5 minutes. Strain the mixture into a glass measuring cup, pressing firmly on the barley to extract as much liquid as possible. You should have about 1 cup. Return the infusion to the saucepan and keep warm.
In a small saucepan, cover the egg with 2 inches of water and bring to a boil. Simmer over low heat until soft-boiled, about 3 minutes. Drain and cover with cold tap water.
In each of 2 large nonstick skillets, melt 6 tablespoons of the butter. Season the chicken breasts with salt and pepper. Add 3 chicken breasts to each skillet, skin side down. Cook over moderately low heat, turning a few times, until the skin is golden and crisp and an instant-read thermometer inserted in the thickest part of the meat registers 155°, about 25 minutes. Transfer the chicken to a work surface and let rest for 10 minutes.
Meanwhile, using a serrated knife, slice off the top 1/4 inch of the egg shell. Using a small spoon, scoop the egg into a blender. Add the warm barley infusion and blend to a foamy sauce. Season with salt.
Remove the chicken breasts from the bone and halve them lengthwise. Arrange the chicken on plates, cut sides up. Ladle the barley foam alongside and serve with roasted asparagus.
Roast the asparagus in a 400° oven for about 8 minutes, until browned and tender.