First consider the four forces on the airplane in the cruise:
⇦ Drag . Thrust ⇨
Drag is a function of the airframe, attitude, and airspeed. If the airplane can fly, then drag is more then offset by thrust. The lift is generated by a component of the forward motion when the flying surfaces convert the horizontal force to vertical force. And finally, the lift force is equal to the force exerted by gravity.
Now let's talk about fuel efficiency. That is how much you can do with a quantity of fuel, and you didn't specify whether you're interested in fuel for time (endurance) or fuel for distance, or something else. But let's discuss the major components of fuel usage:
- Engine motion. The engine is more efficient in terms of power at its higher rated power setting. So, you would think efficiency would be gained by selecting a higher power (assuming you could also climb with less than 100% power).
- Heat. I guess a typical heat engine like a piston engine consumes about 50% of fuel energy to heat (rough guess, it's prob. in the range 35% - 65%). But the heat engine is more powerful the greater the difference between inlet temperature and exhaust temperature. So, the greater time you can spend at higher altitude is probably more efficient in fuel for power.
- Cooling. Sometimes you need extra unburnt fuel for engine cooling. For example during climb, when the engine is at higher power and the airflow is reduced.
Now let's discus climbing. During a climb the aircraft has (in theory) to produce all the power necessary for level flight, plus a little extra for the change in elevation. I would posit, without any evidence, that the fuel spent on the elevation change is insignificant compared to the fuel required to stay aloft. (I will gladly accept a reference or metrics if someone has them). But think about it, if you put the aircraft in an elevator powered by a similar motor, it would take a few drops of fuel to raise a few feet. That's a thousand drops for a thousand feet - it is nothing versus constantly offsetting the the weight constantly as it does in flight.
So what is the more fuel efficient climb procedure? It is the technique using:
- Higher engine power
- Maximizing time at higher altitude; minimizing time at lower altitude
- Balancing the mixture to keep the engine as hot as its design will allow. (you said fuel-efficient. You didn't say cost-efficient. Don't violate your POH!)
In summary, to expend the least fuel, climb as early as possible, as quickly as possible (e.g. Vy) and with minimum acceptable richness.
And as long as we are talking efficiency, please also run the tanks dry at your final cruise altitude. This will maximize the most fuel efficient part of the cruise, when you are carrying less fuel.