Place eggs in a single layer in a wide pan and cover with cold tap water.
Bring just to a boil; reduce heat and simmer for 15 minutes.
Drain; peel eggs under cold running water and place in a bowl of hot water.
In a small pan, melt butter over medium heat.
Add onion and cook, stirring, until soft (about 5 minutes).
Add chile, flour, and milk.
Cook, stirring, until thickened (about 5 more minutes).
In a blender or food processor, whirl avocados until smooth; stir into hot sauce.
Season to taste with salt.Drain eggs, cut in half lengthwise, and arrange on a platter; pour hot sauce over eggs.