1. In a sauté pan over medium-low heat, warm 3 tablespoons of the oil. Add mushrooms, season, cover pan, and cook 10 minutes. Push mushrooms into a ring around sides of pan and add shallots and garlic to center of pan. Season. Cook, covered, until shallots are translucent, about 4 minutes. Sprinkle pan with flour and stir to coat ingredients. When flour has dissolved, stir in vinegar. Increase heat to high and cook, uncovered, until very little liquid is visible. Add stock and cook until reduced to a medium-bodied sauce, 14 to 17 minutes. Chop 1 tablespoon tarragon, add to mushroom sauce, and simmer 1 minute. Season sauce with salt and pepper and keep warm.

2. Heat 2 tablespoons of the oil in a sauté pan over high heat until very hot. Sprinkle skin side of fillets with salt and pepper. Lay fillets skin side down in oil and sauté until skin is brown and crisp, about 3 minutes. Pat dry the top sides of fillets, season, and flip. Cook on second side 2 to 3 minutes. Remove to plate and keep warm. Pour off oil from pan and clean with a bunched paper towel. Return pan to high heat.

3. Add remaining 1 teaspoon oil to pan. While oil heats, place sprouts or other greens in a large bowl and toss with a sprinkling of salt and pepper. Add seasoned shoots to very hot oil and sauté 1 to 2 minutes, until shoots are wilted and slightly charred.

4. Spoon mushroom sauce over each dish. Layer char fillet on the sauce and scatter pea shoots over entire dish. Serve hot.