Is it possible to make a one-off control FET switch circuit?
The circuit you are looking for is called a bistable multivibrator, or flipflop, or latch.  It is a variation of the original multivibrator circuit, which is an oscillator.  You can buy a flipflop as a low-cost integrated circuit in both bipolar and CMOS technologies, or build one with two transistors.
The basic operation is that once the input signal turns on one transistor, the second transistor works to keep the first one turned on after the input signal is removed.
A critical parameter is the amplitude of the input voltage.  Anything over 1 V is easy to deal with.  If you want to stay with MOSFETs, then the input must be greater than 2-3 V for most common devices.  smaller input voltage changes would require an amplifier or comparator stage in front of the latch.
Another part of the design that must be addressed is how to reset the circuit from the latched to unlatched state.  This can be with another signal, a switch, cycling the power to the circuit, etc.
Some of the schematics in this link are of the oscillator circuit, but many are of the latch circuit.
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