As above, the key is fresh eggs. Older they are the messier they get. My advice would be:
– Buy chickens, wait for egg
– Discard live chicken. Reserve in garden for eggs
– big pan, bring about 2″ water
– turn down until bubbles only just breaking the surface
– Crack egg directly into water, no more than 2 for a 12″ pan as it lowers the temp too much
– Cook for 2m45s
– Remove with slotted spoon, drain briefly on tea towel
Source: having ex battery hens. More bloody eggs than you and the neighbourhood can use leads to a lot of poaching experiments.