A quarter of a mile is 400m. That's a long bridge span if it's supported both ends, but as you're only supporting it from one end during construction, that's one hell of a long cantilever!
My first instinct was balanced cantilever construction, to build a single-tower cable-stayed bridge. This construction method is often chosen when building cable-stayed bridges. Basically you build a tower, then add a piece of bridge deck to one side, connect it via cable to the tower, then repeat on the other side to balance it. In your situation, as you'd only have a cantilever on one side, on the other side you'd just be anchoring the tower down to the ground with cables.
However, this becomes a bit problematic when compared to today's technology. The Queensferry Crossing* took the record for the longest freestanding balanced cantilever. And that's for central spans of 650m. Which means that the length of cantilever from each tower was 325m. 441
So the world record for this type of construction is less than your desired distance.
The other option I can think of for building a bridge from one side only is launching it. This is where you build a bridge on land, and then push it out.** This is really best suited for a bridge with multiple short(er) spans, so that for launching over the first span, there's a lot of bridge still on land to act as a counterweight. For use on only one span, you'd need a lot of counterweight! Add in that you're looking at a span right at the top of what's achievable without cables, and it's going to be very heavy and difficult to push.
In conclusion: What you're looking for hasn't been achieved in real life yet. I think it would probably be possible: there's no economic need to build a pedestrian bridge like you've asked for, so no engineers have tried: similar spans are always for heavy vehicles. But in order to build such a bridge you'd need state-of-the-art technology, not simple technology.
* Disclaimer - I was a Design Engineer for this bridge. :D Though I was working on the approach spans rather than the cable stayed ones.
** This is how the approach spans on the Queensferry Crossing were constructed. Super cool videos: no1 and no2. NB Videos are probably only super cool if you like engineering. 933