The 41-year-old US rapper and Cole, 30, are teaming up to get involved in the project which aims to give jobs to young people from less well off backgrounds, as well as donating some profits to charity.
The Chelsea star has faced controversy recently after shooting a work experience student with an air rifle at his team's training ground - but he says this new project isn't about improving his image.
He told BBC Sport: 'I don't do things to try and get publicity or anything like that.
'It was a great way for me to be working with people like Jay-Z and with great entrepreneurs who are going to try and make this restaurant a success.'
The stars were brought together by sports agency NVA Entertainment.
Ashley added: 'I've always loved Jay-Z's music and loved what he's done.
'I know how hard it is to get a job because my mum had chances to get a job and never got them so hopefully I can give something back and this is a great way of doing it.'