Preheat oven to 350 degrees F.
In a dry heavy saucepan, cook sugar over moderately low heat, stirring slowly with a fork (to help sugar melt evenly), until melted and pale golden. Cook caramel, without stirring, swirling pan, until deep golden. Pour caramel into a 13 by 9 by 2-inch glass baking dish and tilt dish to coat bottom. Cool caramel.
Melt butter and cool. In a large bowl, whisk together flour and water until smooth and whisk in butter and remaining flan ingredients until smooth. Pour custard into baking dish and put dish in a larger baking pan. Add enough water to pan to reach halfway up side of dish and bake custard in middle of oven 1 hour, or until set. Remove dish from pan and keep temperature at 350 degrees F. Make the meringue: In a bowl, with an electric mixer beat whites until they just hold soft peaks. Gradually beat in sugar and beat meringue until it holds stiff, glossy peaks.
Spread meringue evenly over custard and bake in middle of oven (not in a water bath) until golden, about 5 minutes. Cool dessert on a rack. Chill dessert until cold, about 4 hours, and up to 1 day. Cut dessert into squares.