Q:

Can a wizard cast the same spell twice simultaneously?

Could a wizard cast the same spell (utility, non-combat [Tiny Servant]) twice at the same time? Using, obviously, both slots, just dual-cast to save a minute or two? I assume the Somatic components for this spell are part of the touching, and there are no material components.
Example:
Blog-hart the Red wishes to re-enact the original Fantasia's Magician's Apprentice. One minute per broom seems too long to him, so he grabs two, and attempts to cast Tiny Servant twice at the same time.

A:

No.
Each spell has a casting time, during which you can perform no other Action. Casting another spell, even the same one, would be another Action. Please see Player's Handbook chapter 9 for how actions work, and chapter 10 for how spellcasting works. 
There are exceptions to the general rules, such as a sorcerer metamagic option called Twinned Spell, and the fighter's Action Surge that can allow a character to do similar things to what you described, but neither apply to your described situation. 

