I'm not sure of the area but you can eat at the Basilica, they have an area where you can order a hot lunch.
There's Busboys and Poets, Chipotle, Potbelly's, Starbucks and &pizza at the Monroe Street Market, which is withing walking distance. If you were coming out of the frontispiece of the Shrine you'd walk to Michigan Ave NE, the main road in front of the Church, at that point make a left and walk for roughly a block, looking for Monroe St. You'll see the restaurants there.
Best choice in the immediate area is the eatery which is in the basilica.
A google search for "brookland cua restaurants" should yield what you need. I've not stopped to eat in the area for some time, thus the lack of personal recommendation.
Hot lunch at the Basilica. Not much of anything for restaurants in the immediate vicinity
Not sure but you can eat a nice lunch at the basilica
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