Remove yeast packet from roll mix. Sprinkle yeast over warm milk in a large mixing bowl; let stand 5 minutes. Stir in eggs and vanilla. Add 1 cup hot roll mix; beat 2 minutes at medium speed with an electric mixer. Stir in remaining hot roll mix with a wooden spoon until thoroughly blended (dough will be soft). Cover and let rise in a warm place (85°), free from drafts, 1 hour or until doubled in bulk.
Punch dough down; let rest 5 minutes, and turn out onto a lightly floured surface. Roll into a 12" x 16" rectangle.
Combine brown sugar and next 4 ingredients; stir until smooth. Spread rum mixture over dough, leaving a 1" margin. Roll up dough, jellyroll fashion, starting at long side; pinch seam to seal. Cut into 16 (1") slices.
Spread Buttered Rum Topping into bottom of a well-greased 10" tube pan. Stand 10 dough slices against outer edge of pan. Stand remaining 6 slices against center tube of pan. Cover and let rise in a warm place, free from drafts, 50 minutes or until doubled in bulk.
Bake at 375° for 20 minutes or until lightly browned. Cool 2 minutes in pan; invert onto a serving platter. Drizzle any remaining glaze over cake.