Thanks everyone for the helpful recommendations on this board. We ate at a lot of markets and had great food, but here are the restaurants we visited in Peru.
Overall the food in Lima was fantastic. The people there clearly care about food and most of the chefs we saw were dedicated to their craft. However, there does seem to be a slight push toward the trendy rather than the tasty which we ran into a couple times. But everyday food there, in the market and local restaurants, was delicious.
Chez Wong (Calle Enrique Leon Garcia 114, lunch only): Solo, Chinese-Peruvian chef starts with beautiful whole flounder and filets it in front of you into the freshest ceviche you've ever had. Unlike some other preparations, this was ceviche in its simplest form: great fish, shallots, lemon juice, salt, pepper. Marinated for less than 2 minutes before reaching our table and served with chopped hot pepper. The restaurant is hidden, hard to find and hard to book, but is worth it just for this dish which is always served. The second course has an option (hot or cold) and we ended up with a simply perfect seafood stir-fry more asian than Peruvian. It was good but in itself not worth the trip, especially at the prices of around 60 soles per dish (both dishes are large enough to split, which most patrons did).
Chinen (Av. Republica de Panamá cdra. 45): We never would have found this, but we tagged along with the family we were staying with in Lima for lunch on the first day to this local hole-in-the-wall. Surprised to see it featured in this Gaston Acurio video, the food was a great introduction to Chifa Peruvian. Don't miss the lomo saltado with the provided hot sauce.
Central Restaurant: This is a top rated, posh restaurant in Miraflores. We wandered in because the restaurant we wanted to go to was closed for an unknown reason and were able to be seated at the bar where you can order from the full menu. We ordered 3 dishes a la carte, the best of which was the charred purple corn octopus appetizer. The arapaima and seabass entrees were good, but at these prices in Lima they were a bit disappointing. The chocolate dessert however was world class - a subtle and interesting combination of . Honestly this visit reminded me why I try to stick more to regional specialities when I travel as compared to ambitious modern / world cuisine. It reminded me somewhat of Vineria de Gualterio Bolivar in Buenos Aires, though much less molecular. It was a meal I felt like could be found in many places around the world with nothing distinctively Peruvian, though there were obvious efforts made to focus on Peruvian ingredients. I would skip this next time for food, but the drinks in the bar were some of the best we had in the country.
Pescados Capitales: One of the most highly recommended restaurants for ceviche in Lima, this too came as a disappointment. Perhaps coming 2 days after Chez Wong was not fair, but we greatly preferred the austere preparations from Chef Wong to the overly sauced, and overly sweet, ceviches here. The fish may have been top notch, but it was hard to tell with what we ordered. This was the biggest disappointment in Lima.
Zig Zag: OK, I'm embarrassed a bit but we stumbled into this tourist trap based on its consistent top marks on Trip Advisor and lack of other info about Arequipa on reliable sites. Perhaps to be expected, it was expensive, the food was forgettable, and the experience was over-the-top. What's the point of serving the meat on a hot volcanic stone? Well it's obviously not a culinary choice and I had to juggle to keep my alpaca at a reasonable doneness. Just look at the photo - it's fun, but if you care about food more than experience skip it.
ChiCha from Gaston Acurio: We didn't have a full meal here, just the peking cuy. Great space, especially the enclosed courtyard. The cuy was delicious if expensive. I'd go back for more.
Cicciolina: We ate here twice, once for breakfast and once for dinner right after our 5 day hike. Both times were great, despite the fact they lacked the local identify I criticized Central for earlier. Oh well-I'm not consistent, I just know these were well enjoyed meals with good wine. I won't fawn over it but if you want yummy, Italian-inspired, gut-filling food, go here. Book ahead as it gets busy.
Greens: Our stomachs just couldn't stand up to the onslaught of street food and we needed something to settle them before our 5 day hike. We came across this restaurant right next to the main square which promised organic, vegetable driven fare with most of its ingredients sourced from its own farm. Sounds risky, but my SF / Berkeley brainwashing made it sound strangely appealing. We had a solid meal here at a somewhat high price. But if you want something simple, light and fresh, I would recommend it.