When I first started making t-shirts I had no idea how to get this effect with my normal sewing machine. I don't have a serger, and my twin needle and I don't see eye to eye (I think the point of having a twin needle is to make two nice parallel lines. My twin needle, on the other hand, thinks its job is to do one nice line of stitching, and skip 9 out of every 10 stitches on the other line. I have tried all sorts of things to solve this problem, but none of them work. None of them. So to punish it, I have put twin needle away in a box to languish sadly and reflect on the error of its ways).
So I made up a way to get the fake-layered look, and I'm pretty pleased with it. The only thing I use which is not totally basic kit is my overcasting foot (Or 'magical brush foot' as I like to call it) but you could do this without- just substitute a zig-zag close to the edge for step 3.
Here's how it goes (with apologies to any metric/imperial purists- I think and work in both, depending on what suits a particular task or measurement. So there are bits of both scattered throughout):
1. Take the normal pattern piece you were going to use to cut out your sleeves and decide where you want to cut it. You want it to look like a short sleeve with a long sleeve underneath. For an inset sleeve, I cut about 1 or 1.5" below the bit where the sleeve seam starts (for a toddler tee). The joining and fake layering is going to mean you lose about an inch of sleeve length so you may want to add an inch to the lower sleeve to compensate for this. I find most of the patterns I use seem to have super long sleeves anyway (or maybe my kids have short arms? But I don't think so) so I just cut my pattern piece into two. What you end up with is two pattern pieces for your sleeve, and two pieces of fabric like this:
2. Pin the pieces RST along the join. I'm a very lazy pinner and these pieces are the same length, so I just pop in a pin at the middle and far end.
3. Use your overcasting foot and a wide zig-zag stitch to overcast the edge. The stitch needs to be reasonably long, too, or your fabric will go all bunchy and wavy. I use a 2.5 length here.
And that's it! The back looks pretty tidy...