Digital or Physical?
Your first step is to "sniff around" to see what they have. What might have holes in it that you can use to dig deeper and find the Donut Preferences Database?
How do you want to start?
Physical Scan
The Department of Donuts has a head office, and a factory for trying out new flavours. You could do some research on those to see if you can learn anything useful. Driving past their building could give us a useful insight. Sometimes there are articles in newspapers or trade magazines that will give us background.
Digital Scan
The Department also have a web site. Their office is probably full of computers. The factory may have industrial control systems or alarms or other things connected to the internet that shouldn't be. You could also research them on social media, or try some stuff via email.
Tools
Hacking is research and exploration. To do that well, you will need a few tools. Learn more about the tools of the trade before you start.