If you allow a coupled inductor, it is possible to do in one transistor. This is a traditional blocking oscillator, with rectifiers handling the power, and zeners controlling off time (usually).
If not, then two transistors are required. One inverts the other, so that, as is necessary for all oscillators, negative and positive feedback can be applied in the circuit. The negative feedback is at DC, so that the amplifiers can always settle into a bias where the loop gain is large, and thus begin oscillating. Positive feedback is at AC, so the oscillation grows over time.
Note that the MOSFET circuit breaks this rule, which is why it is unreliable: there is no feedback mechanism to adjust the first transistor's collector current to around the MOSFET's gate threshold voltage (where the loop will have AC gain, and be able to take off).
Using a BJT, this is okay, because the first transistor is biased such that its collector sits at a bit more than 1*Vbe. It acts as a crude current mirror, where both collector currents are set by Vbe (and some by hFE), and their area and doping ratios.
Note that these circuits cannot possibly incorporate useful control or protective features, like undervoltage lockout, auto-restart in case the oscillation is quenched, or current limiting. (The blocking oscillator can be current-limited by adding a transistor in the emitter circuit, which turns off the main switching transistor when its emitter current rises too high.)
Tim