Heat oil in a deep heavy-bottomed pan until hot.
Combine mashed potatoes, grated potatoes, onions, garlic, and cheese in a bowl.
Mould the mixture into 8 to 10 large flat round cakes.
Dredge each cake in flour, dip in egg, and coat with polenta.
Deep-fry cakes for 4 to 6 minutes until golden-brown and drain on kitchen paper.
Fry shallots in oil for 2 to 3 minutes until softened.
Add wine, bring to a boil, and simmer until mostly evaporated.
Add stock, bring to a boil, and cool for 5 minutes.
Pulse stock mixture, peas, and mint in a food processor into a coarse purée.
Stir crème fraîche into the pea purée.
Serve two potato cakes per plate with a dollop of pea purée.