Your dumb charger is just providing a slightly regulated voltage. It works due to the combined resistances of the charger and battery limiting the current. As the battery voltage gets close to the charger, the voltage potential is less. So with a fixed resistance, the current drops.
I doubt that you can modify your current charger to push 38 A up to 14.4 V. This could only happen if the combined resistances and voltage difference allow sufficient current flow.
Most likely you will need a higher voltage source and then circuitry to make a constant current source to provide a fixed amperage. Shutting it off can be done with a simple comparator circuit. Put a 10k trimmer pot across the battery voltage and run it into a comparator along with voltage from a zener regulator. Then you just adjust the voltage divider from the pot till it trips at the correct battery voltage. Use that to turn off a high current MOSFET or something similar. (Might use the same MOSFET to control the constant current.)
LiIon is usually charged at constant current until a max allowed voltage is reached and then is held at that voltage while current tails off under "control" of chemistry of battery until Ichg = k% of Imax where k% is chosen according to longevity or max energy concerns.50% or 25% of Imax gives longer life. 10% or 5% tail gives max capacity but lower life.
Lowering Vpedesatl by 0.1V greatly assists battery life.
Discharging to higher cutoff voltage aids cycle life.
LiIon also has calendar life and starts self destructing from day one so a lightly used battery still dies.
Best cycle life is achieved by stopping charge when Vpedestal is reached and systen changes from CC to Cv. By monitoring voltage this point can be observed. You could even do a "dumb" system that simply watched delta Vbat and declared constant V when delta fell to zero. Only slightly more than a comparator and an RC delay in one input would achieve that.(While Vin is ramping a delayed vin is lower. When Vin pedestals the delayed Vin almost catches up. An offset voltage is needed to allow comparator towork).
LiIon cells mechanically flex the cell as metallic Lithium is "plated" in and out of the cell*. Cycle life is in large part due to battery beating itself to death mechanically.(This is why LiFePO4 lasts much longer and has lower capacity - the material is held in an Olivine matrix that maintains constant shape as active material is moved in/out BUT it takes up some space. )
*Note: Bill Dubuque has suggested that this sentence would be better replaced with " 'LiIon cells mechanically flex the cell as Lithium ions are intercalated".
The distinction is a finer one than may be apparent. However, it is true to say that if you cut open a LiIon cell you would not usually find metallic Lithium in it. Bill notes that this makes primary Lithium cells, which do contain metallic Lithium, a greater fire hazard than LiIon cells.
If you charge a LiIon cell with excessive voltage metallic Lithium will be 'plated out' and "vent with flames" mode usually occurs at about the same time.
Charge to CV level as often and as soon as possible.
If charging all the way their "disconnect message" is a sign of bad ethos. They are probably trying to minimise the risk of fire without telling you.
For longest storage life (as opposed to long life in regular use) storing at a lower voltage than Vmax is in order. Probably at about 3.6 V and only about 30% state of charge. The various Mars Rovers use LiIon batteries and have a design life of about 8000 cycles - but charge to about 3.6 - 3.7 V maximum.
8000 / 365(~=) ~= 22 Terran years.
Best Answer
As long as the battery is nearly empty, your charger produces its nominal current and runs constantly at a frequency that is likely around 100 kHz > 15...20 kHz = your audible range.
Once the battery is nearly full, the charger goes into burst mode. This means that it switches at 100 kHz, stops switching entirely, and then produces some more cycles at 100 kHz (i.e. the next burst), stops again, ... (repeat).
What you hear is not the main switching frequency, but the frequency of the bursts. You can think of this like an amplitude modulation between 0% and 100% with an inaudible carrier of 100 kHz and a signal of the burst frequency, likely of some kHz and in the audible range.
The reason this is done is that you are able to greatly reduce the standby power consumption of a charger using burst mode. (A nearly full battery looks to the charger like "almost standby".)
For further reading, you can search for keywords like "burst mode" and "switch mode power supply".
The components producing the noise are the transformer, and maybe the EMI filter chokes.
It could also be the case that the charging circuit inside the cell phone modulates its current consumption and thereby imposes an audible modulation onto the switch-mode power supply inside of the charger device.