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Spring and early summer are usually considered the best time to raise chicks due to a variety of factors (increased fertility of rooster and eggs, better nutrition from having fresh greens to eat, increasing day length, etc).
You'll need a broody hen and fertile eggs to pull this off. With a rooster in the flock, it sounds like you have the fertile eggs part of the equation covered. The breeds you mentioned are not likely to go broody, but there are exceptions. There's really no way to make a hen "go broody". It totally depends on the hen. For a hen who is already genetically predisposed to go broody, the increasing day length of spring usually brings on broodiness (although some hens will go broody anytime of year). A hen who is broody will stay on the nest all day and night, only getting off for 10-15 minutes per day to eat, drink and poo. If a hen is already predisposed to go broody, it may help to let eggs accumulate in the nest in the spring.
If you have a hen go broody, you can decide which eggs you want her to incubate. For example, you can choose eggs from your best laying hens to put under her. There's much more information about broody hens in the Learning Center.
It's absolutely delightful to watch a hen with her chicks, and it's so much less work as she does all of the work for you!