Sprinkle the unflavored gelatin over 1/4 cup of dark rum and allow to soften for 5 minutes.
Meanwhile, stir together 1 1/4 cups of whipping cream, brown sugar, and espresso powder in a saucepan over medium-high heat. Bring to a simmer, stirring until brown sugar has dissolved. Remove from heat, then whisk in gelatin mixture until dissolved.
Whisk in the coconut milk, vanilla extract, and sour cream until smooth. Evenly divide the mixture between eight 3/4 cup custard cups or molds, cover each with plastic wrap, and chill at least 4 hours to overnight.
Bring 3/4 cup whipping cream and corn syrup to a simmer over medium-high heat. Once simmering, remove from heat, and stir in the chocolate until melted and smooth, about 2 minutes. Stir in 1 tablespoon of rum and set aside.
To serve, run a knife around the edge of each mold, then set each cup into a shallow bowl of hot water for 10 seconds to loosen. Invert the mold over a serving plate and remove the panna cotta. Spoon chocolate sauce around each panna cotta and garnish with a sprig of mint.