Whatever you do don't do it in midsummer - not a good time to repot or train. H.Chestnuts don't make great bonsai because of the growth habit of the very large leaves, but you can still enjoy growing it and learning while you do. What kind of maples do you have? Next (early) spring would be the right time to do work - when the buds have fattened up and are just about to open... depending on where you live, that could be late March to April some time (why not put your climate zone in your profile?). There's a lot more to do besides miniaturizing - you need to have the right soil mix(es), know about light and how to water (if at all, again depending on where you are... ahem :-), etc. etc. The trees must live outdoors for life of course, they're anything but indoor trees, so it's important to understand them and the climate they now live in so you can take proper care of them. And, you won't like this part, but they really should be in the ground for a few years... that way you get the fastest natural growth that you then cut back at appropriate times so a fatter trunk develops (not just a stick in a pot), and the branches are 'shrunk' with bonsai techniques.

Knowing your zone is very important in giving correct care instructions. Most as suggested would be more concerned with thickening of the trunk. The quickest route would be ground growing. If it could winter your area safely. Not sure I would focus on a deep pot more so than a wide pot. Bonsai you want lateral roots, the deeper the pot to my minds thinking is forcing the roots to grow downward and not producing the lateral roots by not having a wide pot. Many who are growing out saplings and such will build a grow box...or even use colanders from a kitchen store. Which are wide yet not so deep. But you have many years before you begin to do work on this one. But...having it thrive under your care and developing it into a prebonsai...can be quite rewarding. Though...if your looking for hands on fast...depending on your budget...and what you want to work with. Looking into winterization and proper care to keep it happy and healthy during that time will also help going into choosing something that will be in the long run...a very rewarding experience. Looking into native trees/shrubs might be something to look at as well. Though...tropical bonsai can be raised with success. Some are more finicky...it pays to know what to expect said plant to do...and care it needs.

I wish you tons of success and joy in this new hobby you have taken up. Some pick up cheap plants at a local nursery...some prefer prebonsai material. I have had success online with New England Bonsai myself...picked up a rosemary bonsai from them.

My suggestion also would be...see if there is a local bonsai group in your area...you might be surprised but that also is a great way to acquire bonsai material along with hands on work...as well as knowledgeable folk who have something in common as you.

Welcome to a fun hobby...don't let this little sapling hinder your desire of working hands on with something...this one just needs several years to develop to be able to be workable. But, that doesn't mean it won't be rewarding. You can always pick up a more hands on workable tree if you desire working one right away.

Root Trimming / Tap Root Cutting is something else you're going to want to read up on

Best of luck

Here is a link that might be useful: Some Root Info.

Looked a bit closer to your photo...your tree's leaves are turning brown. What is the care you are currently giving it?

Soil, watering schedule...you mention it is on a balcony. Is it in direct full sun? What part of the day if so...and how many hours.

Not sure if you have sunburned leaves...or what is going on. A close up of those leaves might be helpful.