Any diet that is adequate for keeping canaries healthy and breeding will also be a good diet for the Gray Singing Finches, since they are closely related, with both classified in the genus Serinus. My own basic daily diet for them is white proso millet, canary seed, hard-boiled and mashed egg, crushed eggshell, and some greens each day. They are particularly fond of dark greens such as kale, rape, collards, and broccoli leaves. Rapeseed grows beautiful blue-green leaves that any birds in this family dearly love. Everyone who raises any of the serins should have a small plot of rapeseed planted, so that you can give each bird one small, tender leaf of these greens every day.
Eggs are cheap, highly nutritious, available anywhere, and they are easy to prepare.
Also, eggs contain a nearly perfect balance of vitamins, minerals, and fats. Once any bird learns that this is a food item, the bird will eat it regularly and willingly. Cool the eggs in cold water, and then peel the shell off. You ll find that the shell peels off very easily, because the egg absorbs the water as it cools, and this additional water keeps the shell membrane from sticking to the shell and makes the egg much easier to shell. Once the shell is off, dry the egg gently in a paper towel to remove the excess moisture.