My husband made this for me for Christmas. He changed the size a bit but overall stuck to the plans. He used Jacobean stain by Varathane and 3 coats of matte poly by Varathane as well. He used an HVLP sprayer to apply the poly and I think that made a huge difference in the smoothness.
Matte finish by varathane for poly. 3 coats with HVLP sprayer.