Visiting NYC over Christmas - where should I eat on Christmas day?

Asking for help! My girlfriend and I are visiting New York over Christmas (we're from London) and I've been tasked with finding somewhere special for us to eat on Christmas day. I know that Thanksgiving is the bigger event than Christmas Day in the US, but in the UK 25th of December is the king.

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I'd like to find somewhere to eat that is really special and memorable -- it can be expensive but ideally not crazy expensive. I don't mind spending a lot of money, but what we'd love is to not feel guilty about splashing out on one too many bottles of wine and to leave notably fatter and drunker than when we arrived.

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We're staying near Bryant Park, but can travel. I know we're not too far away from Eleven Madison Park, but I imagine getting a reservation there will be difficult and it's also going to be really expensive and potentially quite stuffy? Looking for somewhere not too formal, with amazing food/atmosphere etc. that will do our waistlines and cholesterol levels proud, and will be memorable for like $500 or less for two people.

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Would love to know where people would recommend.