In the bowl of an electric mixer on medium speed or using a sturdy whisk, whip the eggs, brandy, and ½ cup of the sugar together until smooth, about 2 minutes. Add the cocoa powder and cookie crumbs, and whisk until smooth. Gradually add the milk in two stages, stirring until smooth between additions to avoid lumps. Cover and refrigerate the batter overnight.
Pour the remaining 1 cup of sugar into a heavy sauté pan (not nonstick). Add 2 tablespoons of warm tap water, stirring until the sugar is evenly moist. Cook over medium heat until the sugar begins to turn light amber in color, 5 to 6 minutes. If the top crystallizes and hardens, stir the crystals back into the liquid sugar until melted. When the sugar is evenly golden, reduce the heat to medium, stand back, and gradually pour in ¼ cup of warm water; it will bubble and steam. Cook and stir the caramel just enough for it to remelt and become a pourable syrup, about 30 seconds.
Immediately pour the caramel syrup into a metal or ceramic 9-inch round cake pan or other 8-cup baking dish (avoid glass, as the hot caramel may shatter it). The bottom of the pan or dish should be covered with a ¼-inch-thick layer of caramel. Refrigerate the caramel until it is completely cooled (it should feel tacky), at least 1 hour or up to 4 hours.
Heat the oven to 300ºF. Bring a 1½-quart-size pot of water to a boil. Briefly stir the refrigerated chocolate mixture; then pour it into the pan over the caramel. Set the pan into a larger, deeper pan (such as a roasting pan) and pour boiling water into the deeper pan to fill about halfway up the cake pan. Bake until slightly puffed yet wobbly in the center like a firm custard, 40 to 50 minutes. Remove the pan from the water and let it cool on a rack for 15 minutes. Refrigerate the cake in the pan until completely cooled, at least 2 hours or up to 4 days.
To unmold the cake, run a wet knife around its edge, and top it with a large plate or platter. Invert both pan and plate, shake the pan to loosen the cake, and then lift off the pan. Cut the cake into eight wedges and serve each with some liquid caramel from the plate.
Tip: After making the caramel, immediately soak the pan and utensils in hot tap water to melt the caramelized sugar. It makes cleanup much easier!