Hi @khatriy,
Ideally, you would remove the top piece of the chair, then drill out the dowel that has snapped, add a new dowel, then glue it back together with some wood glue such as this Gorilla Wood Glue.
Unfortunately, this might not be a simple job if the rest of the timbers are still well connected. If the top can come off, I would do this, but if it is just one snapped dowel, I would use a simpler approach.
A simple repair would be to simply add some wood glue in the opening and on the dowel, then use some long clamps, like these Craftright 600mm 2 Piece Quick Action Clamp, over the top and under the base of the seat, to hold the pieces together while the glue sets.
Make sure you wipe away any glue that squeezes out while it is still wet.
Let me know what you think and if you have further questions, please don't hesitate to ask.
Jacob
