No breed does both eggs and meat well, except for some strains of Pekin ducks and Appleyard ducks. The breeds that do both eggs and meat aren't all that good at either one.
If you are looking for a self-sustaining flock after TSHTF, I suggest that you keep 2 flocks. One flock for eggs and the other for meat. Get Leghorns for eggs and either Dorking or True Cornish for your meat flock. The Dorkings and Cornish will lay plenty of eggs to be self-sustaining, but not enough eggs for your family to eat an omelet every morning.
The leghorns lay eggs like crazy and they are good to eat, but they don't have good feed conversion for the meat part of their production and the dressed birds are tiny. Their feed conversion for eggs is phenomenal. You will get a lot of eggs for your feed, just not much meat.