For indoor training, I don't need to eat a regular breakfast. Generally, I'll grab half a banana or a Lara Bar. Have something, a combo of carbs of protein, but not too much.
However, for a regular outdoor ride (I often do a short ride as early as 5 AM some days), I have a Lara Bar, or some plain Greek yogurt with fruit. Then, I'll have some more food when I get back. It's kind of like splitting my breakfast in half. If I am going on a longer ride, a bit later in the day, then I have my regular breakfast, usually with the addition of some carbs I don't usually eat (like spelt toast with almond butter or a whole wheat bagel half).
I knew someone who came to spin class at 5:45 AM without eating anything. That was fine for that, but one day she went out on an early AM ride and ended up passing out. A policewoman found her lying next to her bike! So, put something in your stomach, but experiment with what works.