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If a monk uses a staff and has the Dual Wielder feat, can they use a 1d6 attack for one hand then 1d6 attack bonus for the other, plus gaining +1 to AC for holding a melee weapon in each hand?

Seems a bit much to otherwise require 2 staves.

The Monk has to wield two staves to get the benefits you list

The Dual Wielder feat specifies (emphasis mine):

You gain a +1 bonus to AC while you are wielding a separate melee weapon in each hand.

So while the monk is using one staff, they don't gain this benefit, nor can one use Two-Weapon Fighting with a single weapon wielded in two hands (emphasis mine).

When you take the Attack action and attack with a light melee weapon that you're holding in one hand, you can use a bonus action to attack with a different light melee weapon that you're holding in the other hand.

With two staves, however, you can certainly benefit as you describe as Dual Wielding removes the requirement for light weapons:

You can use two-weapon fighting even when the one-handed melee weapons you are wielding aren't light.

Is this too strong?

Using two staves in this way is no stronger (by itself) than any other dual-wielding combination with monk weapons, so there should be no issue.

It seems like your question is coming from a slight misquote; you left out an important word. The Dual Wielder feat doesn't say "a melee weapon in each hand", and it's very clear about what it requires:

You gain a +1 bonus to AC while you are wielding a separate melee weapon in each hand.

It specifically requires a separate weapon, not one weapon that you have both hands on. A single quarterstaff, no matter how you use it, is only a single weapon, and doesn't qualify.

I'm not sure where you're getting two 1d6 attacks with a staff. Are you suggesting that a staff held in two hands would also count as two weapons for Two Weapon Fighting (it doesn't), or are you talking about the monk's Martial Arts ability to "make one unarmed strike as a bonus action" after an attack action (which would be an unarmed strike, not an attack with the staff)?

Just to be clear, unarmed strikes aren't weapons, so they don't apply towards the "separate melee weapon" requirement. A weapon plus an empty hand doesn't work; two empty hands plus Martial Arts doesn't work; a staff held in two hands doesn't work; a weapon in hand plus a dancing sword doesn't work. You need two actual weapons in your actual hands.

I'm not sure why you said 'two staves is a bit much' -- if you were going to do this with a monk, you'd probably want to use two smaller weapons, like nunchaku (clubs), short swords, or similar. Dual quarterstaves might be a bit silly, yes, but that's not the only option, or even the most obvious one.

For more details on why unarmed strikes aren't weapons, see the Sage Advice Compendium question regarding Stunning Strike and the errata document section marked Weapons (p. 149), as well as the errata for Melee Attacks (p. 195), which says in part:

Instead of using a weapon to make a melee weapon attack, you can use an unarmed strike[...]

(Since an unarmed strike is "instead of" a weapon, it clearly isn't one.)