Preheat the oven to 300 F. Grease 6 6-ounce ramekins with butter or nonstick cooking spray.
Heat the milk and cinnamon stick in a medium pot over medium-low heat. Bring to a simmer then remove from heat, cover with top, and set aside to steep while preparing the custard.
To prepare the caramel, place 3/4 cup of sugar, 1/3 cup water and the lime juice in a small saucepan over medium heat. Do not stir with a spoon, move the pan around to mix everything together. When the sugar turns to a deep caramel color, remove from the heat and evenly pour the caramel into the ramekins. Place the ramekins in a deep baking dish.
Beat the eggs, egg yolks, and remaining 1/4 cup sugar with an electric mixer until pale and creamy. Remove the cinnamon stick from the milk and discard. Slowly add the milk and instant coffee into the egg mixture. Whisk or stir until fully combined.
Pass the mixture through a sieve into a bowl. Evenly pour the custard into the ramekins leaving 1/4-inch of room at the top.
Fill the baking dish with boiling water so the water comes halfway up the sides of ramekins.
Bake for 1 1/2 to 2 hours, or until the flan has set but is still a little wobbly in the center. Cool to room temperature and then place in the refrigerator until cold.
To serve, run a knife around the edge of each flan and then turn out onto plates. Garnish with coffee beans.