Cut 2 1 1/2-inch-thick rounds lengthwise from center of cabbage; reserve curved ends for another use. Cut each round in half lengthwise, keeping stem intact. Place cabbage steaks in a shallow dish. Drizzle with vinegar and 2 tablespoons of the oil. Let stand for 30 minutes, turning after 15 minutes.
Place eggs in a small saucepan, covered with cold water by 2 inches; bring to a boil over high heat. Remove saucepan from heat; cover and let stand for 9 minutes. Drain. Plunge eggs into a bowl of ice water; let stand until cold, about 15 minutes.
Put 1 tablespoon of the remaining oil in a small skillet with shallot. Turn heat to medium-low and cook until sizzling. Let sizzle for 30 seconds; transfer to a medium bowl to cool. Peel and chop eggs. Place eggs, lemon juice, capers, cornichons, tarragon, and remaining 2 tablespoons oil in bowl with shallot; fold together. Taste and season with salt and pepper.
Heat a grill pan over medium heat for 5 minutes (or preheat grill). Coat with cooking spray. Coat cabbage steaks with cooking spray. Grill cabbage, turning occasionally, until lightly charred and tender, 10 to 12 minutes. Remove from heat. Sprinkle evenly with salt and pepper. Place 1 cabbage steak on each of 4 plates; serve with sauce.