It's "without breaking a few eggs." One egg doesn't make much of an omelette.
You use it when you are making changes that involve a disruption of the present order.
Maybe a company is doing a major restructuring of its operations and things will be chaotic for a while.
Or there is major work happening on the roads or bridges. The eventual new bridge will be worth the effort, but while the bridge is being worked on, the traffic will be worse than if they did nothing.
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