I recently reminisced of last autumn, when I assisted in "breaking" aquifers for the single-pick challenge - reawakened thread last year. I checked the wiki but apparently noone has done a writeup of it. I'd quite like to do it myself but probably won't have time before June since I'm trying to graduate nowish. So meanwhile, you can go read
this post and the next post, which has some refinements to Stochasty's method (which builds off earlier work by Merendel and others earlier in the thread, and the cave-in methods known by the wiki and the general community). Pages 12-15 have most of the relevant posts (don't think there's almost anything after, may be some before).
Basically, as long as you have at least 2 dry soil layers it's possible to pierce a 2-layer (soil) aquifer with nothing else but 1 pick and miner. And it's a lot easier and less hassle than using pumps (the double-slit method). This is done by doing a mini-chicken run (a term used in the aquifer article on the wiki; a pure chicken run is trying to dig down into the cavern before your miner drowns) in order to drain the upper aquifer level into the lower.
Your miner needs to dig out the stairs in the 2nd layer before the 1st fills with water, so you need to train their mining skill: it has to be something like 8 to have a (10%+ish) chance at success, but embarking with mining 5 and digging the preparations will get you there, or very close, if you use only 1 miner as recommended. And if you're going for a 1-tile pierce, you get 12 attempts, so it's ok to give it a shot before it's guaranteed... 12 skill or more makes it a pretty sure thing. A 3x3 pierce would have 16 attempts, a 5x3 18, 5x5 20 etc. I think, but to be honest you can just do infinite attempts off to the sides of the basic layout and drain to there if you really need to.
This method can be done, and was in fact developed for, reanimating terrifying/otherwise very dangerous biomes (you can wall yourself off underground as soon as you get your dwarves underground, preferably with the wagon wood so you can make an axe later), or a freezing biome (you don't have to open up the aquifer to the surface, so it doesn't freeze), all you need is 2 levels of soil before the aquifer. The 1-layer aquifer pierce is definitely doable if the surface isn't deadly, but I don't think we ever figured out how to pierce a 1-layer aquifer with only 2 layers of soil above while remaining completely safely bunkered. There's still tricks to minimize the exposure time, of course.
If your surface is safe and your aquifer is only 1 deep you can just do the cave-in outlined in the wiki, but I'd like to add that instead of just channeling a 5x5 area of the aquifer, you can also just channel the 3x3 in the middle. Then dig down stairs in the ring around that, and up* stairs into the aquifer. The stairs won't stop the cave-in unless if they're not in the cave-in area, as my screenshots show. This way you won't have to build floors to access the plug after the cave-in.
*(or up/down like I actually did back in September, but up is safer since you won't flood your fort if you dig out the space underneath those stairs)
P.S. Redwood Elf: If by "2 levels down" you mean 1 dry soil and then aquifer, and the aquifer turns out to be 2 soil layers, a modified (surface-exposed) version could work. First just train your miner somewhere to sufficient skill (10+). This training should include a ring (7x7 for a 1-tile pierce) of down stairs on the surface. It can** also include channeling the ring just inside that from the surface (except for 1 tile which should just be dug on the level below, so you leave 1 tile of floor for support), and another ring of stairs, this time up/down (again, 7x7 for a 1-tile pierce) in the 1 dry soil layer. Then make a mini chicken run to pierce a drain from A1 (Aquifer 1, counting from the surface) to A2. Once you've succeeded at this, you can dig out the top aquifer layer and channel a hole in A2, and cave-in a plug from the dry soil layer into A2.
edit:
**The chances of mini chicken run success are improved if you start it with the miner in a shaft, so they don't do the initial dig diagonally from 1 tile to the side, since then they don't have to move said 1 tile over before moving down and digging into A2. Hence you should only do the up/down stairs in the corners + every 2nd tile (on each mini chicken run attempt spot) in the dry soil layer if you want to optimize this aspect, or the miner will likely stand on a neighbouring up/downstair or on an up ramp left from the channel instead of right above what he's digging.
P.S. Now I started thinking about whether a miner can dig a channel on the level below if they're standing on down stairs, which would make the 2-layer aquifer pierce doable even with 0 mining skill. Probably not.