Birds, fish, and everything inbetween.
A problem of legendary proportion
It is no secret that there exists an imbalance between a single boss monster and a party of players. At its core, this comes down to Action Economy. Taking many actions is simply more effective and versatile than only taking one. Even with Legendary Resistance and Legendary Actions, a single boss monster is often easier than multiple enemies. Adding minions or liutenants help, but is not always the right solution for the story.
This system aims to close that gap, while remaining intuitive and fully compatible with the rest of the rules. Not all legendary creatures will use this, but it is another tool in the DM's toolbox.
Legendary Initiative
Legendary creatures with this trait roll initiative multiple times, going at all resulting initiatives. To make it less swingy and dependent on the initial dice rolls, at least one player character must take their turn between each turn the creature takes. This is resolved when initiative is rolled by adjusting the rolls of the legendary creatures (preferring adjusting downwards). The monster's turns also can't bookend the initiative order, since that would also make it go twice in a row.
By taking additional turns, Legendary Actions are no longer needed to shore up the action economy, especially since the creature also gains more Reactions per round. Recharge actions are also rolled every turn, possibly increasing their frequency. Some creatures may have additional restrictions on their actions, such as only being able to cast a spell once per round.
Since the creature is able to make multiple saving throws per round against crowd control effects, and each individual turn lost is less catastrophic to its overall action economy, this also removes the need for Legendary Resistance.
The exact number of turns the creature takes is a tuning variable. A goblin boss, intended for a tier 1 party, might take one additional turn, while an adult dragon, intended for a tier 3 party, might take three or even four. The number of turns could also be affected by temporary effects. Perhaps it gains an extra turn while Bloodied, or dealing a specific damage type removes one of its turns. At its most extreme, the creature could go after each one of the player's turns.
Certain actions of the creature can also take more than one action. For example, a dragon charging its breath weapon (indicating its direction) before releasing it on its next turn is very cinematic, and gives players a chance at tactical counterplay, either by moving out of the area (or helping allies move out), or by potentially incapacitating the dragon.