Something to note is with things like the LHC, they are not accelerating single pairs of particles. They are accelerating a whole bunch (billions) and waiting for some to invariably collide.
For the double slit experiment, you can control an electron gun by varying the current. Electing guns are related to the technology that powers CRT screens. You basically heat metal which causes electrons to break free (at a rate that is proportional to temperature) and then you use magnetic fields to move them at a desired rate. Go low enough with the current and the electron release is super slow/rare.
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