Add 3¼ cups coconut milk, coffee, brown sugar, cocoa powder and cinnamon to medium saucepan over medium heat. Heat coconut-mocha mixture, whisking occasionally until cocoa powder and sugar dissolve, about 5 minutes. Transfer coconut-mocha mixture and rum, if using, to medium pitcher. Cover and chill until very cold.

To make martinis, pour remaining coconut milk into small rimmed plate; transfer flaked coconut to separate small plate. Dip rim of martini glass into coconut milk to moisten, and then dip into sweetened flaked coconut to thoroughly coat. Repeat with remaining glasses. Fill prepared glasses with chilled coconut-mocha mixture. Serve.