Mosfet or bjt choice for low voltage joule thief application
Indeed 0.5 V is too small to do anything useful with a BJT.

For MOSFETs the threshold voltage \$V_t\$ matters, the ones with the lowest \$V_t\$ I can find have \$V_t\$ between 0.3 V and 1.0 V, for example the Si2334DS.

That 0.3 V sounds good however that is the minimum value, the manufacturer guarantees that \$V_t\$ is always 0.3 V or higher (but less than 1.0 V). This does not mean that you can buy a MOSFET which has a \$V_t\$ of 0.3 V. Typically the \$V_t\$ will be around 0.65 V I guess, so that will not be more usable than a BJT.

There are solutions which can work on a voltage of 0.5 V but these use a dedicated power harvesting IC, for example the BQ25570 but do not that even this IC needs at least 0.6 V to start up. Once started it can work down to 0.1 V at its input.

Do note that a BQ25570 solution is significantly more complex than your "Joule thief" idea.

You really need a bit more voltage 0.5 V to do anything useful at all. Can't you use two solar panels in series?