Whisk together yolks and granulated sugar until mixture is thick and pale (about 2 minutes). Stir in coffee extract. Sift together flour and next 3 ingredients; gently stir into yolk mixture until just blended. Stir in butter and milk until just blended.
Whisk egg whites until stiff peaks form. Fold one-third egg whites into yolk mixture. Fold in remaining egg whites.
Pour into a well-buttered 10-cup Bundt pan.
Bake at 350 for 50 minutes or until a wooden pick inserted in center comes out clean. Cool in pan on a wire rack 10 minutes. (Do not remove from pan.)
Meanwhile, bring brown sugar and brewed coffee to a boil in a medium saucepan. Reduce heat to low, and simmer, stirring occasionally, 20 minutes.
Remove from heat; stir in rum. Reserve 2 cups syrup. Return pan to stove top; cook over medium-low heat, stirring occasionally, 15 minutes or until slightly thickened.
Pierce cake 10 to 15 times using a skewer.
Pour reserved 2 cups thin syrup over cake.
Let stand 15 minutes. Invert cake onto a serving platter. Spoon desired amount of thickened syrup over cake. Reserve remaining syrup for another use (such as topping pancakes).
*The rum is optional. Replace it with an additional 1/2 cup brewed coffee, if desired.
