$$\epsilon \ddot y + (1+x)\dot y = 0 \;\;\;\;\;\; y(0) = a \;\;\; y(1) = b$$
How do I find the "leading order composite approximation" of this BVP? I'm not totally sure I'm approaching this right, but I thought to find the 'outer solution' by looking just at $(1+x)\dot y = 0$ which I think means $y$ is a constant. I guess that's possible, but I don't have much experience at all here and it feels wrong that I'm not getting a function of $x$. Is this the right way to start this? How do I move forward? What's "composite" about this?
Your first order outer solution is correct, simply set $\epsilon = 0$ and $y = $ constant.
Now you need to look for an inner solution. Nothing seems fishy near the center of the interval (assuming the problem is on [0,1]), so checking the endpoints for a boundary layer is usually the next step. I haven't worked through it, but usually one scales to boundary layer coordinates, meaning use $\xi = \frac{x}{\epsilon^a},$ where $a$ is to be determined by balancing. If one concludes that the layer is not at $x=0,$ then the scaled coordinate $\xi = \frac{x-1}{\epsilon^a}$ would be the scaling to look at the boundary layer near $x = 1.$
Once you have obtained the inner layer solution, matching the two will provide any remaining constants (that isn't always true, but for most textbook problems it is).