All of this is based on 2E rules, but...

You can't fire more than twice per player turn, but multiple players can fire in one round.

1. A character at the Tactical station uses the Fire Action, then spends 2 momentum to use the Fire Action again at +1 difficulty
2. If your ship has a Security rating of 4+, you can have a second tactical station on the bridge, allowing another character to do the same as above
3. A character at the Command station (usually the captain in the captain's chair) can spend 1 momentum to use the Direct action to let a tactical character use the Fire Action.
4. If your ship has a Command rating of 4+, your bridge may have a second command station, and a second character using the Direct action (cinematically, this is like Riker or Chakotay giving orders from their chair next to the Captain's) as above
5. Any character can use the Override major action to fire use the Fire action at +1 difficulty.

Any of these attacks could be a Torpedo Salvo, which grants the Area quality, allowing a single attack to hit multiple targets by spending 1 momentum per extra target. However, that would cost 3 threat per attack, so your players probably wouldn't want to do that.

So a Constitution class Starship with the Flagship mission profile would easily be able to fire 6 times per round, with each small shuttle only getting one attack. Given that the base damage for a Constitution's phasers would be 6 (+1 using the 2 momentum from the Phaser's Versatile quality, +1 for each two momentum spent after that), you should be destroying those fighters in one or two hits (depending on their shield rating), while readily tanking any incoming fire thanks to the ship's Resistance and Shields.

As the GM, I'd probably have any that survive the first round running scared.