Vernick Tavern is excellent (small plates--fine dining type of cooking) and walking distance in Rittenhouse Square after the Barnes (but make reservations). If you don't mind taking a cab, I would also recommend Zahav (Society Hill-Israeli) or Zoo Kee Do (Norther Liberties, Portuguese)
+1 for everything Jean said! Here is another example of the Matyson Sunday dinner menu.
November 25, 2012
Lobster & Squash Coconut Curry
mango chutney, peanut, thai basil
lobster & shrimp sausage, broccoli rabe, pine nuts
Butter Poached Lobster
potato, fennel, pernod, sourdough
Apple Cranberry Strudel
caramel ice cream
$35 prix fixe does not include tax, beverage or gratuity.
The Matyson tasting menu usually has adventurous dishes, sometimes very adventurous like when they did an all-offal menu. It's also walking distance from the Barnes. The menu will be posted a few days before Sunday here: http://www.matyson.com/tasting.html
Personally I think Branzino and Estia are about as far from adventurous as you will find in Center City, though the daily flown in fresh fish at Estia is very nice (and expensive). The Sunday menus at Amis and Bistrot are also not very exciting though Bistrot is a good deal.
It would help if you could expand on what you like to eat (are you looking for exotic ingredients or just preparations that aren't mundane?), do you want to walk from Barnes or planning or drive or cab, and general price range.
Barry - thanks for catching the part about adventurous eaters. With regard to Matyson you are confusing the Sunday menu and the tasting menu. They have a different menu available just on Sundays. It usually is 10 dollars cheaper and not necessarily the same as the weeks tasting menu which is from Monday to Thursday. For example tonight's menu is:
MATYSON SUNDAY DINNER MENU
Sunday, November 18, 2012
Cold Smoked Turkey Breast
Cranberry, celery root, kale, brie
Meatballs, dark meat, crispy skin
Cauliflower puree, rye spaetzle, brussels sprouts, maple mustard
Gingersnap Ice Cream Sandwich
$35 Prix fixe does not include tax, beverage, or gratuity
The link for the Sunday menu is: http://www.matyson.com/sundays.html
To the original poster.. if you could be more specific about what you mean by adventurous eating (OFAL ethnic sushi etc) we probably could provide some more suggestions.
Four restaurants that come to mind that have specials on Sunday are Amis and the Dandelion Pub. Amis, off of Broad Street has a family style homemade spaghetti and meatball special. Dandelion Pub, in the Rittenhouse neighborhood has Roast beef and Yorkshire pudding. Both comfort foods will go well after having done the marathon that is the Barnes. Matyson, also in the Rittenhouse area, and a BYO, has Sunday special meal that varies each Sunday and is worth considering. Price point is about $35.00. Lastly, Bistro La Minette off of South Street has a great Sunday pix fixe meal special. There are of course plenty of others but I just wanted to highlight special deals available on Sundays for dinner. With the exception of Dandelion Pub, each of these are available beginning at 5pm.