I designed and made this circuit but it produces only 360V when it should be producing at least +1000V
As this is a standard 50/60Hz transformer, you cannot drive it at 1kHz. Laminated iron cores work up to roughly 100Hz. Above, the eddy current losses are enormous.

As this isn't a flyback transformer, you have to supply AC without a DC offset to it to avoid saturation. You need a H-bridge instead of a single transistor and commutation diode.

As that H-bridge supplies a rectangular voltage to the transformer instead of a sine, you have to use a 1.6 times higher frequency to account for the higher flux. Fortunately, 96Hz is within the limits of a laminated iron core.