It is possible, but in fact it is often very difficult. With your Greek asylum status, you are only allowed to work in Greece. And your Greek travel document only allows you to travel to most other European countries for a visit of maximum three months, but you cannot work there. To be able to work in another European country, you must obtain a working visa. Every country has different requirements for it. So you first need to check if you are able to meet the requirements for a working visa in the country where you want to work in.
For example some countries require that you already have a signed job contract, proof of an accommodation where you will stay, or proof of skills of the country's language. If you have higher qualifications, like a university degree, your chances might increase. When you are sure you meet the criteria and you have all the required documents, you can apply for an appointment with the embassy or consulate of the country you want to work in. The processing time for a working visa is normally between 6 and 12 weeks after all documents are handed over. If you are able to get a working permit, it is in most cases valid for one year, with the possibility to renew it