Bring barley and water to a boil in a small pot. Reduce heat and simmer, covered, until barley is tender and water is absorbed, about 50 minutes. Let cool.
Heat oil in a large skillet over medium-high heat. Season chicken with salt and pepper and saute until golden and cooked through, 5 to 6 minutes per side. Transfer to a plate to let cool slightly. Shred.
Heat pan with chicken juices over medium and add onion, carrots, and mushrooms. Saute until tender, 6 to 8 minutes. Deglaze pan with wine.
Whisk flour into chicken stock. Add to pan. Bring mixture to a boil and cook until slightly thickened. Stir in barley and shredded chicken. Let cool slightly.
Heat oven to 425 degrees. Using a 1 1/2-cup ovenproof dish or ramekin as a guide, cut phyllo to make tops (keep covered with damp towel). Spoon mixture into four ramekins. Brush each cut sheet with oil and place on top of filled ramekin. Cut vents into tops. Sprinkle with sea salt.
Transfer to rimmed baking sheet and bake until golden and bubbling, about 30 minutes.