For a more casual but very tasty lunch also try Bertha's Cafe on 32nd St and Indian School. Great soup salad and sandwich place, and nice atmosphere for a lunch deli place. I like Miracle Mile as well, for a chain, they do pretty good pastrami. Another great neighborhood place is on 44th and Camelback, Taylor's Cafe. Perfect for a friendly lunch with awide selection of bruch and lunch items.
2822 N 32nd St Ste 1, Phoenix, AZ 85008
It's a bit outside of the Biltmore area, but Cherryblossom Noodle Cafe sounds like it may fit the bill. They have very good Japanese dishes, and a surprisingly delicious selection of Italian entrées as well. The atmosphere is casual but nice; it looks like... well, like a casual Italian restaurant with Japanese touches.
Across the street from Biltmore Fashion Park in the Esplanade shopping center/office complex is a terrific little French cafe called Au Petit Four. It's more airy than cozy, but certainly a worthwhile stop. It's rather relaxing to sit and sip a glass of wine while watching all those poor harried businessmen scurry about like so many ants.
Duck and Decanter is fairly close too... it's been there for ages. Something of an upscale deli. I haven't been, sadly. I keep getting drawn into Miracle Mile whenever I'm in the area.
Andy's mention of Coup des Tartes is quite possibly the winner. I didn't even know they were open for lunch until I read his post.
Cherryblossom Noodle Cafe
914 E Camelback Rd Ste 1, Phoenix, AZ 85014
Biltmore Fashion Park
2502 E Camelback Rd # 216, Phoenix, AZ
I really don't think you'll find the Vig too noisy expect perhaps during the peak happy hour. I've been three times and have never had a problem with the noise. The quietest portion of the restaurant is probably in back near the bocce court.
I also agree with Rubee about the Parlor. Very nice place.
On the grounds of the Biltmore resort itself, the casual restaurant was recently rechristened as "Franks & Albert's" and is also a good lunch choice.
I've been enjoying The Parlor recently. Great pizzas (faves are the Funghi with wild mushrooms and goat cheese, and the Salsiccia with grilled radicchio, sage and sausage made by Schreiner's using a Parlor family recipe), pasta made in-house, bake their own bread, etc. Love the cocktails using herbs from their garden (a favorite is the Black Velvet Sage with gin, blackberries and lime juice) and that they source local ingredients. Nice comfortable vibe too, with friendly service.
The Parlor Pizzeria
1916 E Camelback Rd, Phoenix, AZ 85016
Miracle Mile Deli at 20th Street just south of Camelback is a good lunch choice. Great deli sandwiches (The Straw or the New Yorker are excellent) in a very bright, cheery atmosphere. Noise level is very subdued and service is quick and efficient (counter line service).
Miracle Mile Deli
1949 E Camelback Rd Ste 160, Phoenix, AZ 85016