Nope. Can't do that with any activity, including casting.
You have to do an activity all at once. In an encounter, this means you must complete it during your turn, but outside of encounters, activities can take minutes, hours, or days. If an activity gets interrupted in an encounter, you lose all the actions you committed to it. For an activity that occurs outside of encounter mode, you usually lose the time you put in, but no additional time beyond that.
Oh, found more specific text:
You Cast a Spell you know or have prepared. Casting a Spell is a special activity that takes a variable number of actions depending on the spell, as listed in each spell’s stat block. You can spend those actions in any order you wish, provided you do so consecutively on a single turn. As soon as all spellcasting actions are complete, the spell effect occurs.
Is it possible to split the spell casting over into a second round?
For example; As a third action in a round, to start casting Fireball, then complete it as the first action of the next round.
Or with the casting of Magic Missile spend all the actions for two rounds creating 6 magic missiles. I know this is not a sensible action, I'mn just asking if it's possible
by the rules you can't.
But in reality you really should be able to do that.
You risk many chances to be interupted in multiround casting.