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Egg yolk is a fine binder for pigment, having been used for centuries.
The process is fairly simple
- Crack an egg
- Separate the yolk from the white
- Dry the yolk
- Puncture the yolk sack
- Drain into a cup
- Add about the same volume of water
- This is your binder
Egg tempera and casein should be painted onto a rigid surface like a panel. You can prepare that panel traditionally with rabbitskin glue gesso, as described here. You could also mount paper to a panel to provide a rigid but absorbent surface.
Read this post for some guidance on pigments.