2 places I recommend
These are the two restaurants and ice cream shops in Vicenza I have eaten at myself and would recommend.
Where a section has more than one place, they are numbered from my favourite downwards. 
marks the ones not to be missed.
Don't forget you can take the map with you
A traditional Vicentine osteria on Strada Pozzetto, on the edge of the old centre, in the Rigodanza family's hands for generations and listed as one of the city's historic shops. The room is all wood panelling, and it's best known for baccalà alla vicentina, the city's salt cod slow-cooked with onions, anchovies, milk and cheese. As usual, I opted for pasta, which felt like a home-cooked dish. Simple but very satisfying local dishes.
Address: Strada Pozzetto 10, 36100 Vicenza VI, Italy
A small, modern gelateria on Corso Fogazzaro in the centre of Vicenza that calls itself "il gelato gentile", with the gelato kept in lidded steel pozzetti and flavours like saffron with black pepper and raisins alongside the classics. Gambero Rosso gave it their prize for the best savoury gelato in the country. They also make cakes and biscuits. It was one of the best gelatos I've ever had in Italy. The standout flavour was the sesame and toasted peanuts.
Address: Corso Antonio Fogazzaro 56, 36100 Vicenza VI, Italy
