I have 3 EE's; 2 I bought full grown and one "americana" chick (roo) that i raised to adulthood. All mine are tame, even tamer than the bantam cochin and the orpington hen that i raised from chickhood. I have had 3 roos altogether, and i am at a loss as to whether they can be pet material. They are great pets without hens, but once you get them..... Mine will jump in my lap, but then other times will not hesitate to spur me. He is affectionate but over protective of his girls, with roos it's best to buy adults then you know for sure how they will act. But he is very gentle with chicks i brought home, after supervised visitation until i trusted him he would feed them and stand between them and the hens who were not so nice (hens don't like newcomers in general). He also finds food for my girls and makes them nests and sings about it. My hens jump right into my lap too. One lays a light blue egg and the other lays a pink one.
Another gentle breed is favorolles, they are also low on the pecking order. They also have the beard, and have extra toes! I think mine are mixed with them possibly. In this breed there is the salmon, and i believe blue wheaten color.
If you want to be friends feed them bananas; they go nuts! just not everyday. I have not found it to slow down egg production at all. Takes care of the shyness in a jiffy. Chicken candy. Other breeds i have tried start out as friendly chicks, but after 3-4 months get really wild all of a sudden.