Todd English's Food Hall (The Plaza Food Hall) in the basement of the Plaza Hotel on 59th and 5th Avenue (entrance on 59th). There's lots of options and stations (pasta, ramen, pizza, sushi, salads, raw bar, etc.). Inside you'll also find very New York food vendors like Luke's Lobster and No. 7 Sub. If you like real Japanese sushi, Sushi Den (Sushiden NY) on 49th and Madison is also a great choice, though I would venture over to Sushi Yasuda (Sushi Yasuda Ltd)on 43rd and 3rd if you're a serious sushi lover.

Betony is fabulous and right in your desired area

Momofuku Má Pêche is a great choice. Spacious, good atmosphere, good food.

Molyvos is a great greek restaurant, can sit inside or outside. Rouge Tomate has a business lunch special $32pp one appetizer, entree and dessert but their normal menu is larger and not toooo pricey. The Capital Grille is right over there too if you're going for a steakhouse vibe.  Maze Restaurant by Gordon Ramsay is always a satisfying meal as well.

Marea at 240 Central Park S is fantastic for lunch. Ambiance is very business-y but the food is beyond! Pricey but the lunch deal makes it very doable. 

Serafina Broadway at The Dream Hotel is always a safe bet and good for work lunches. Delicious Italian food. There are multiple locations around NYC and one of them is a few blocks over from where you said - 210 West 55th St. Eataly is another great option though it's a bit more of a walk, also Italian, very novel setup. And afterwards for dessert since you're in the area hit up Momofuku Milk Bar  if you're still in Midtown or Doughnut Plant if you're in the Flatiron. 

TAO can get loud but offers great food. Aspen Social Club has some great drink selections. 

Marea would be a great choice.  Excellent food and not too far from your intersection.

My favorite in that area is Fig & Olive. The prices are pretty reasonable and it has a great vibe. Mostly Mediterranean/Italian style food and a big menu to choose from. Definitely make a reservation, though, as it gets very crowded. 

Circo at 120 W 55th st has a great prefix lunch for 29$. best in NYC.

You are at an odd location - Tiffany & Co., St. Patricks Cathedral, Saks Fifth Avenue. The only places that I visualize are the nearby hotels like Omni Berkshire Place, The Peninsula New York and The Westin New York Grand Central. A local may know of a few good little bistros on the side streets. You'll need a reservation anywhere you go.

I would vote for the Russian Tea Room