As long as your asylum case is ongoing, you cannot change to a residence permit. Once you finish your Ausbildung, you can then work while still having the Aufenthaltsgestattung. You have to get permission from the Ausländerbehörde for each job you do while having an Aufenthaltsgestattung.
Alternatively, once your Ausbildung is completed, you can cancel your asylum application and apply for a residence permit according to paragraph 18a of the Aufenthaltsgesetz. This permit is if you have completed an Ausbildung and found a job in that sector. I would advise you only to cancel your asylum application once your Ausbildung is complete, you have a job, and have sought additional advice from an expert organisation (e.g. your local refugee council ‘Flüchtlingsrat’) or lawyer. This can help you decide whether it is worth your while to remain in the asylum process while working or to change to a work permit according to 18a.