I am designing a cold-blooded dragon about the size of a house cat with a wingspan of 3 metres from tip to tip. It will climb on top of rocks at noon when there is ample sunlight overhead. I need it to take off vertically and remain several meters high in the air for at least ten minutes to escape from predators. What biological mechanism does the cold-blooded dragon need in order to perform such a feat?
All flying animals are warm blooded to some degree. Pterosaurs, birds, and bats are all warm-blooded. Insects are even warm-blooded compared to other arthropods, they are technically cold-blooded on the ground but use their massive wing muscles (which occupy most of their thorax) and to rev up their metabolism before they fly, and are temporarily warm-blooded while in flight. Other insects that spend long periods of time in flight (looking at you, bees and wasps) have additional adaptations to keep the body at the proper temperature.
Taking off vertically is also really energy-demanding because you are going straight up at a high angle of attack rather than at an angle or letting gravity, thermals, or a running start help you. Even among birds, only a few species can even take off vertically and in general it is taxing.
By definition, dragons are reptiles, all of which are normally cold-blooded. Laying eggs is also a basic trait of reptiles world-wide.
For your cat-dragons, to avoid predators by taking off vertically, they would need highly adapted reflexes tuned to detecting that predator, and strong legs that would launch them high enough into the air to allow them to deploy their wings and begin flying without dropping back into the claws of their predators.
You may want to reconsider your wing engineering for your beastie as well as the larger wings will take more muscles and time to create lift for your creature to fly. Smaller wings that burst into action as the creature leaps would be more effective in achieving flight.
I'm not sure they would be limited to 10-minutes of fly time, you would need to define why they could only fly for short periods of time based on further refinement of your physiology. You are encouraged to read Anne McCaffrey's Pern series.