Preheat oven to 350°.
Combine 3/4 cup sugar, cocoa, flour, and salt in a small saucepan. Add water; stir well with a whisk. Bring to a simmer over medium heat; cook 2 minutes, stirring constantly. Place chopped chocolate in a large bowl. Pour hot cocoa mixture over chocolate; stir until chocolate melts. Stir in the rum and vanilla.
Place 1/2 cup sugar, eggs, and egg white in a bowl; beat with a mixer at high speed for 6 minutes. Gently fold the egg mixture into chocolate mixture.
Divide chocolate mixture evenly among 10 (4-ounce) ramekins coated with cooking spray. Place ramekins in a 13 x 9-inch baking pan; add hot water to pan to a depth of 1 inch. Bake at 350° for 25 minutes or until puffy and set. Serve warm.