Season fish with salt and pepper. Melt butter in a large frying pan over medium-high heat. Cook fish until browned underneath, about 3 minutes. Transfer fish to a plate, browned side up.
Add carrots to pan along with broth, wine, vinegar, peppercorns, bay leaf, and tarragon sprigs and bring to a boil. Reduce heat and simmer until carrots are tender, 8 to 9 minutes.
Meanwhile, combine lemon zest and minced tarragon and spread over browned side of fish. Slide fish, browned side up, and any accumulated juices into pan. Add tomatoes and simmer until fish is just cooked through, 2 to 6 minutes depending on its thickness.
Remove and discard bay leaf and tarragon sprigs. Divide fish among 4 wide, shallow bowls, then ladle vegetables and nage into bowls. Drizzle each bowl with 1/2 tsp. lemon juice and 1 tsp. oil.