It doesn't allow you to, but if you're not gaining anything between turns, there's no consequence to doing it that way.
Gaining between turns, or being forced to discard between turns, or discarding reactions, or being deck-inspected, or changing your mind about which card you were going to play first after seeing what opponents do and playing sift before draw.
Now there's an interesting puzzle: A complete list of every possible situation that can make it relevant whether you shuffle with a small or empty deck while other players are playing, or whether you have to wait for your turn to avoid corrupting the process. I'll enumerate what I can think of.
-- Gaining: Junkers (curses/ruins/coppers), gifters like Messenger/Embassy
-- Drawing: Council Room, Lost City
-- Topdecking: Ghost Ship, Haunted Castle
-- Inspecting: Spy/Oracle/Scrying Pool
-- Sifting: Oasis/Inn/Warehouse (If you change your mind about the order after watching what others do, Even playing Market/Oasis instead of Oasis/Market could change whether the shuffle triggers before or after you've discarded the card to Oasis.)
-- Reactions: Any reaction that gains, draws or discards can matter, even if played on your turn like self-trashing/Market Square, depending on the order you play them, and the decision about the order can follow what other players do.
-- Changing your mind: Having written two "changing your mind" items, I just had one of those "oh, no" moments. Shuffling is technically relevant at all times and should never be done outside of your turn. I'm going to have a hard time selling that to the playgroup, though. Scenario: Your hand is Smithy/C/C/C/E. You have 1 card left in your deck and it's not your turn. Your opponent draws some cards and plays treasure and you realize that you're light on economy. You decide NOT to play the Smithy and buy a Silver now so that it doesn't miss the shuffle. Therefore, shuffling before your turn would have changed the outcome.
Yeah, I'm definitely going to have a hard time selling that one, especially since early shuffling only hurts the shuffler. But let's flip it around. Shuffler has draw in hand, late in the game, but decides to buy VP after doing a mental point count. But they've already pre-shuffled. Now their newly purchased VP card misses the shuffle.