The main issue for me is that my interests are a bit too broad. Should I learn web dev from my job by day, then try to implement an OS in my free time by night? I know I can't learn everything, but yet it feels like there's so much I really _do_ need to know to be a competent developer.
There really isn't anyone on the planet who is truly a "full stack developer" and competent from the front-end all the way down to the OS and bare metal. No one.
Most developers stick to a particular layer, and are only familiar with adjacent layers to the extent of being able to have arguments with those developers. A 5x-ish developer is competent in those adjacent layers as well. A 10x developer either has mastery of their chosen layer, adds additional competency layers, adds competencies on the connecting tech between chosen layers (ie. not just front-end + backend, but networking too), has relevant domain competencies, or some combination.
So, just pick an entry point, learn as much of it as you want, check out whatever is adjacent, follow your interests, and periodically evaluate whether you need/want to change course or dig deeper into technologies, tools, domains, etc. wherever you happen to be. Even if you end up wandering quite far from where you started, the experience and knowledge you gain on the way isn't likely to be "wasted", unless it is mastery of some in-house tech or tool that literally no other organization uses.
Separately, "the stack" is inherently polyglot, and even aggressively pruning the selection wherever possible and eliding many formats, tools, and protocols still gives you an absolutely minimal set like JavaScript, HTML, CSS, HTTPS, TCP/IP, SQL, C/C++, Assembly, & VHDL.
You value practicality as well as personal interest/fulfillment. Either pick one and start there or pick both and consider the options there. If you want to be practical, learn the most marketable skill that you're at least somewhat interested in: Java, AWS, Docker, Javascript are all incredibly marketable. If you want to go for interest, just pick the thing you're most interested in. You could also try and juggle two deep dives, at the cost of more time and less depth, but maybe that works for you.
Don't worry about becoming a senior dev in a semester. Pick something and focus on it. That could be an AWS certification, building something in Java, etc. Once you get through that, then decide your next step.
If you want to spend a whole winter break + semester (which is a very valuable amount of time), is there anything else you're interested in that would also line up with your career interests? That might be a win-win for the academic exercise and practical benefit