They definitely can overwinter. And, in your climate that should be possible.

Couple of things:

• they don’t grow well over the cold months and go into a sort of semi-dormancy.
• pruning them heavily before spring seems mostly detrimental. they may try to push new growth which will suffer, and seem to benefit from a cloak of leaves/dead stuff.
• they don’t tend to do well in subsequent years. the center of the stem doesn’t become woody and instead starts to rot out. (I’ve done it several times. They have problems.)

If you end up with a plant with some viable shoots come January/February and you’re not expecting any frost, rooting some cuttings or layering from the original plant can give you viable starts for spring planting. (This probably makes the most sense in your scenario.)

If you plan on starting from seed, most places that sell starts get theirs going first week of January. They have a small seed and take a while, especially in cool weather, to get good enough size for transplant.

The other option is seeding in place, which preserves the native taproot damaged in the transplanting process. This can result in more vigorous plants, but can be challenging in terms of weeding and managing pests. (With you growing on a balcony, the benefits/challenges here might be minimal.)

It depends on where you are, and what your winters are like. Where we are they’re already dead and gone, but friends who are closer to the equator still have theirs producing. There isn’t information in your post to give an accurate answer, but if you provided your location there would be more to base an answer on

Lots of informative other comments. Just wanted to say I love your balcony and cute doggo.