This reader used an "acid base stain" 40 years ago to stain some birch doors, Michael Dresdner has some ideas on how to make a similar finish today.
Wood Stain Colors on Birch - Olympic
How To Stain Wood - A Step By Step Guide - Wood Finishes Direct
Custom Craftsman Shaker-Style Interior Door
Coloring Glue - Woodworking, Blog, Videos, Plans
How to Build a Two Sided Barn Door - Sawdust Sisters
How would I replace these wood panels and existing glass in antique pocket doors with my own pieces? : r/StainedGlass
The Ultimate Guide to Staining Plywood - A Butterfly House
Avoid Color Mistakes and Learn How to Fix a Blotchy Stain - FineWoodworking
How to Update Oak or Wood Cabinets WITHOUT a Drop of Paint (PART 2) - Kylie M Interiors