Can I Quicken a cantrip to cast it twice in a turn and also use Twinned Spell on both castings?

Question

I'm currently building a monk sorcerer who uses melee spell attacks, notably the cantrip Shocking Grasp. Can I use the Quickened Spell Metamagic option to cast the spell as a bonus action and then use the Twinned Spell Metamagic option to attack an additional enemy, then use Shocking Grasp again as my action and use Twinned Spell again to attack both enemies a second time?

Answer 1

No, you can not.

The rules for Metamagic contain this line

You can use only one Metamagic option on a spell when you cast it, unless otherwise noted.

Neither the Quickened Spell nor the Twinned Spell metamagic have exceptions allowing them to be used with other metamagics, so in the absence of other features allowing you to use multiple metamagics on the same spell you can not used Twinned Spell with the same spell that you used Quickened Spell on. You can cast one Shocking Grasp as a Twinned Spell and one as a Quickened Spell though (although this is probably an inefficient use of your sorcery points).

FYI if you want to know what an exception to the metamagic rule would look like, the Empowered Spell metamagic contain this sentence:

You can use Empowered Spell even if you have already used a different Metamagic option during the casting of the spell.

Answer 2

You can use one Metamagic option or the other for a single casting of a spell, but not both at the same time.

The Metamagic section of the Basic Rules states (emphasis mine):

At 3rd level, you gain the ability to twist your spells to suit your needs. You gain two of the following Metamagic options of your choice. You gain another one at 10th and 17th level.

You can use only one Metamagic option on a spell when you cast it, unless otherwise noted.

In this case, we can look at the Metamagic option Empowered Spell as an example, which explicitly states you can use additional Metamagic options. Emphasis again mine:

Empowered Spell

When you roll damage for a spell, you can spend 1 sorcery point to reroll a number of the damage dice up to your Charisma modifier (minimum of one). You must use the new rolls.

You can use Empowered Spell even if you have already used a different Metamagic option during the casting of the spell.

Let us now look at the description of Twinned Spell:

Twinned Spell

When you cast a spell that targets only one creature and doesn’t have a range of self, you can spend a number of sorcery points equal to the spell’s level to target a second creature in range with the same spell (1 sorcery point if the spell is a cantrip).

To be eligible, a spell must be incapable of targeting more than one creature at the spell’s current level. [...]

and of Quickened Spell:

Quickened Spell

When you cast a spell that has a casting time of 1 action, you can spend 2 sorcery points to change the casting time to 1 bonus action for this casting.

Note that neither of these Metamagic options state that you can "stack" them with each other, like the Empowered Spell option explicitly states.

Shocking Grasp targets one creature (and isn't otherwise disqualified, see here), so it is eligible to be Twinned. It is also eligible to be Quickened, but neither option allows you to use both on a single casting of the spell.