Charging a high voltage capacitor with power supply. How exactly?
The example power supply you show has a current limit and a voltage limit: it will limit is output to whichever limit is encountered first.
Into a short-circuit, or a discharged capacitor, it will supply 0 to 10 miliamperes as set by one of the knobs on the end panel.
If you get a supply like that, but that can be set to 4000V instead of the 5000V lower limit of the example device, it will do what you want.
I see your correction from 1500uF to 1.5uF that's now of the approximate scale of the capacitors present in microwave ovens, (slightly larger and slightly higher voltage)