Preheat oven to 325 deg F.
Boil milk and then cool it till its warm. Add sugar to the milk and stir to dissolve it.
In a blender, blend eggs and vanilla extract together. Give it 2-3 pulses till its combined thoroughly. Strain it through fine mesh.
Add eggs to the milk and stir well.
In a small saucepan, add sugar and water and give it a stir. Heat it in medium flame till its starts bubbling.
Add lime juice and swirl the syrup. Keep it bubbling till starts to attain a golden brown color. Switch off gas and swirl the caramel to check consistency.
Prepare six small mounds or ramekins and pour little caramel in each mould distributing it equally.
Place the moulds/ramekins in a cake pan or tray with its height almost equal to the moulds.
Pour boiling water in the pan taking care it doesn't splash in the custard. Adjust the water so that it reaches midway to the height of the moulds.
Bake in oven for approximately 1 hour (depends on temperature of individual ovens, so preferably keep checking on the custard after 30 minutes).
To check for doneness, toothpick when inserted in the custard should come out clean. Custard should have a slight wobble.
Remove from oven and let it rest for 5 minutes. With a knife, loosen out the custard from the edges and place a plate over the mould. Flip it to dislodge the custard from the mould
Serve chilled or warm.