I'm similarly one month into a new senior development role. Although I am familiar with the tech stack, the system's functionality, deployment environment, and points of interface with other systems are fairly complex. The other devs on the team will answer specific questions but are pretty stingy with their time when it comes to something like an overview of the system. I could (and have) read through a fair amount of the code, but IMO, that is very inefficient and potentially misleading way (you might be able to see what is happening but not necessarily why, something that is important for senior devs to understand) to come up to speed on it. Oddly enough, the organization has a very structured onboarding process for everything except coming up to speed on the specific details of their product's functionality. Frankly, learning everything about their expense reimbursement policy and the health insurance website by day two is nice, but not nearly as useful as coming up to speed on how their product works. I'm finding it pretty frustrating.
Both here and in previous roles, I start with the data model and work my way backwards. If you don't understand the data, it's nearly impossible to understand what the system does. In previous roles, I'd also browse through the production environment (e.g., AWS) and look at the monitoring pages for the different components (databases, EC2/ECS instances, queues, etc.) in order to get both a sense of the topology of the production environment and where bottlenecks might exist. In previous roles, I've been able to make some significant improvements early on as a result of that. In my current role, utilization is Earlier in my career, I had a team lead who hired me into a role on a complex system and he scheduled one hour per day with me for a full month to walk through various aspects of the system. Obviously that was a pretty significant time commitment, but I think it paid off for him in that I came up to speed pretty quickly and was quite productive. That always stuck with me and when I became a team lead, I always budgeted a significant amount of my time to bring new members of my team up to speed. I'm pretty frustrated in my current situation as an unproductive senior developer due to unfamiliarity with the functionality of the system and confused as to how its beneficial to the company to operate this way since I am rather well-compensated. I think it's quite penny-wise and pound-foolish for organizations to hire developers and invest as little time as possible in bringing them up to speed.