I'll second the Oliver's and Los Agaves recommendations in Sheridan.
I love the Waldorf A'Story in Story, Wyo., for sandwiches at lunch. Still haven't been for dinner, but I hope to make it for Indian-taco night someday.
The Branding Iron in Dayton, Wyo., is famous for its pies. I also enjoyed its salad bar, especially the macaroni salad and pea salad, and its fried chicken.
Back to Sheridan: The Silver Spur Cafe (breakfast and lunch only) makes a heck of a white gravy that more than makes up for the so-so biscuits and/or pre-frozen chicken-fried steak.
The Big Horns cover a lot of territory! But, I can vouch for Sheridan. There is an absolute gem of a restaurant called Oliver's on south Main St. Food is superlative. Ingredients are fresh. Excellent, reasonably priced wine list - Matt's wife runs a wine and cheese shop next door. Highly recommended.
Also on Main Street is the Java Moon, coffee and deli. Nice place for lunch and the only outdoor seating in town.
Los Agaves is a wonderful Mexican restaurant.
I love Sheridan. I used to live in Worland and we fished and camped in the Big Horns. My sis and broher-in-law run the Rams Horn Cafe in Worland. Love the drive from Ten Sleep to Sheridan...and back around thru Shell and Greybull. I was just wondering if anyone "elsewhere in America" knows about eating spots in or near the Big Horns. I'll tell my brother about Oliver's in Sheridan. They motorcycle over the mountain lots to go to Sheridan. (I live in CA now). P.S. My brother-in-law makes the best Greek Salad I have ever had..at the Rams Horn Cafe. Well, he is Greek!