If its going to be home made then adding whey or Grow! or something is not on track.
If you are willing to use a powdered supplement which would probably be cheaper and more effective in the long run, just mix a whey blend or Grow! or something with whole milk. You can then chuck some oils in. Might taste like shit with olive oil and stuff in it, be aware.
If you want purely homemade, first of all, I don't recommend using raw eggs, and cooked eggs in a shake will be rank.
A good clean bulking blend that won't taste like the slime that oozes out of rubbish (garbage) bags in kitchens, would be as follows :
2 cups of Milk preferably whole and maybe organic / filtered.
Equiv of 2 eggs worth of pasteurised liquid egg. You can get in in a container from supermarket.
4 Tablespoons skimmed milk powder
2 tablespoons natural yoghurt (Greek yoghurt is higher in protein btw)
1 tablespoon lecithin granules
1 ripe banana
1 cup of strawberries or other berries.
Other fruit could be blueberries or similar. You can buy bags of frozen berries all year round, just make sure you don't do what I do and destroy your blender so make sure they are defrosted.
That lot above should give about 55-60 g protein and the calorific value is variable depending on the milk and yog used.
My advice is buy some Grow! or something and mix whole milk with it, in fact you could mix something similar in a bowl with yoghurt and oatmeal that's what I've been known to do. Particularly if you don't like the feeling of liquid sloshing round your guts.