Position a rack in the middle of the oven and preheat to 375 degrees F. Butter the bottom and sides of four 6- or 8-ounce ramekins.
Melt the butter in a medium saucepan over medium heat. Add the flour and salt and cook, stirring, about 1 minute. Add 3/4 cup water, bring to a simmer and cook, whisking constantly, until the mixture is as thick as pudding, about 3 minutes. Remove from the heat and stir in the cheese, fines herbes and whole eggs. Transfer to a medium bowl and stir until slightly cool.
Beat the egg white in another medium bowl until soft peaks form, about 2 minutes. Gently fold the whites into the cheese mixture until there are only a few streaks.
Put the prepared ramekins on a baking dish, divide the batter evenly among them and smooth the tops with a spoon. Bake until golden and puffed about 1 inch above the lips of the ramekins, 15 to 20 minutes. Serve and enjoy right away.
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