Climb Performance
In simple terms climb performance for shallow climb angles is calculated as follows:
Rate of Climb = [(thrust - drag) x TAS)]/weight
where TAS and ROC have the same units of speed (kt, m/s, fps, or whatever you choose)
or, put another way, approximates to:
TAS x [ (thrust/weight) - (drag/lift) ]
You set the best lift/drag ratio by flying at the speed for minimum drag. This will vary slightly with weight but does not change with thrust. Therefore the "best climb speed" is what it is. You fly it and you get the best climb rate because you have the best available L/D ratio. If you increase speed to try to get a better climb rate, the drag increases and thus you don't get the gain in climb rate you might expect.
The biggest factor in the equation is Thrust/Weight ratio. If you are at MTOW and it is hot outside (ISA + 17) then with the best will in the world your T/W ratio will be poor, so your climb rate will be low. To get a better rate all you can do is take off at a lower weight. Even the best pilot in the world will be constrained by the equations above.
The only way you could get around this, would be to do a series of diving accelerations, each interspersed with zoom climbs, converting speed to height in several steps. It can be more efficient but I expect ATC might have something to say about it in busy airspace!