This quote is essentially a metaphor for persistence and exploration. The "vein" Thoreau refers to could be understood as a path or a pursuit, be it a career, a hobby, a study, or any other endeavor. The idea is to continue working on this path until it is "exhausted", meaning until you have extracted all possible knowledge, skill, or benefit from it.

Alternatively, this path might lead you to a "broader one". This suggests that through dedication and persistence in one area, you might discover a larger, more promising opportunity. It’s the concept of one door leading to another, bigger one. It emphasizes the importance of dedication and the willingness to delve deep into one’s pursuits, as it can lead to unexpected and greater opportunities.

Applying this to today’s world, it could be interpreted as a call to specialize and become an expert in a particular field. In an era where information is abundant and easily accessible, it’s easy to skim the surface of many topics without truly understanding any of them. Thoreau’s quote suggests that we should resist this temptation and instead focus on mastering one area.

In terms of personal development, the quote advocates for patience and persistence. It encourages us to stick with our passions and interests, even when progress seems slow or the end goal is unclear. It assures us that this dedication will either lead to mastery or open up new, unforeseen paths. It’s a reminder that growth and discovery often come from depth, not breadth, of knowledge and experience.