You'll need a passport if you have to clear immigration in Kyiv. If you fly a low cost carrier such as Ryanair, or otherwise need to change between terminals D and F at KBP, then you'll need to go through Ukraine immigration, and this requires a passport. Italian citizens do not need a visa for entering Ukraine though.
If both of your flight segments are on Ukraine International, then you should be able to transit without going through immigration in Kyiv as both of these flights use Terminal D.
The UIA web site explains that you don't need the passport if you remain in the international transit area:
The passengers-citizens of EEA Member State, who travel from EEA member state only on UIA flights (operating carrier is PS) with the connection through Boryspil International Airport (from INTERNATIONAL to INTERNATIONAL FLIGHTS) and who do not require visa for the further travel are accepted for transportation providing that they have a valid ID card. The presentation of a passport for this category of passengers is not mandatory.
Since you are physically exiting the EU it is best to have a passport, even if you might not need it, in case of any emergency or change of plans.