In a large bowl, whisk together egg and egg yolks. While whisking add the sugar slowly until fully dissolved. Add vanilla, heavy cream, milk, and salt and continue to whisk. In order to get rid of air bubbles in mixture, transfer to large bowl through a fine strainer.
In the bottom of 6 (1/2 cup) ramekins (small ceramic custard cups), place about 1 1/2 teaspoons of caramel. When all 6 ramekins have caramel at the bottom, brush each bottom and side with melted butter to secure flan and yet allow it to be removed easily. Fill each ramekin with the flan mixture, SLOWLY to avoid air bubbles, until about 2/3 full.
Preheat the oven to 350 degrees F.
Place almost full ramekins in shallow baking pan and then fill the pan with boiling water. The hot water bath will allow the flan cups to cook evenly in the oven. Right before placing pan in oven, top each ramekin with a tablespoon of hot espresso. Bake for 40 minutes.
When removing pan from oven, allow the ramekins to remain cooling in their original water bath, overnight if necessary.
Before serving, place ramekins in another shallow pan of warm water to loosen the custard. Run a thin knife around the inside edge of cup to further loosen, place a plate on top of ramekin and flip onto plate to serve.
In preheated saucepan over medium heat, pour in water and bring to boil. Add sugar and stir until caramel brown (about 4 minutes). Remove from heat and allow to cool for about 2 to 3 minutes.