This may be a better question for the woodworking stack than for home improvement. They'll almost certainly give you better answers. But speaking as a half-trained woodworker:
Most wood glues, applied properly and clamped until dry (which admittedly is a bit tricky in this case), are at least as strong as the wood; often stronger. So simply gluing it back together is not out of the question. Though given the direction the grain goes, it may break somewhere else if snagged again.
I'd be tempted to drive a few pins through the piece and into the frame after the glue had fully cured, if I could find a good place to set them, to help strengthen it against being pulled forward in the future. But that's partially a pinning gun looking for an excuse to be used. Brads or trimnails would work too, but being thicker they'd want to be drilled for first and then countersunk with a nail set, and ideally then have the space over them filled in with putty/stain (or at least furniture repair wax in a shade to match the wood they're driven into) plus a bit of finish (shellac's traditional for touch-up) to match the sheen of the current varnish.
To make the crack less visible, you could stain the "show" edges before reassembling, or use a stainable glue and stain afterward. I'd lean toward the former, though I've found stain markers, applied carefully, can hide a multitude of sins.
(Stain marker, matching fill-in crayon, and a small bottle of shellac are often sold as a furniture scratch repair kit. You'd want to match the dark color, not the gold.)