Using a LED with an ESP8266 module
I'm not quite sure why you categorise the new circuit as better and more efficient. Where to connect the controller is a good question. I would suggest that you don't.

Your original circuit is almost the standard method for this 3W use case.  You could add a 100k resistor between the gate and ground. This gives you some confidence of the start up state of the MOSFET as it may have accumulated charge. Since it worked, we can infer that the necessary MOSFET gate drive is &lt; 3 V. Your incorporation of the 2.2 Ohm resistor prevents thermal runaway in most cases. 

A 1A constant current supply is by no means more efficient in terms of design and parts.  Especially if you only have 3V to play with.  So why mess with it?