Mix all-purpose flour, grated parmesan, salt, and pepper in a shallow bowl.
Press each chicken medallion into the flour mixture, ensuring full coverage before shaking off excess.
Preheat a heavy-bottomed pan over medium heat, then melt butter with olive oil.
Once hot, place chicken medallions in the pan, leaving space between each.
Cook for about 3 minutes on the first side, then flip and cook for another 2-3 minutes until golden brown and cooked through.
Transfer to a plate and repeat with remaining chicken, adding more butter and oil as needed.
Lower the heat and sauté shallots in the same pan until soft.
Stir in minced garlic and cook until fragrant.
Pour in chicken stock, scraping up browned bits from the pan, then bring to a simmer.
Add gnocchi and cook for 5 minutes, stirring occasionally until tender and the stock reduces.
Reduce heat and mix in butter, heavy cream, lemon juice, lemon zest, capers, and salt.
Stir to combine and melt the butter.
Return chicken to the pan, coating it in the sauce, and let it warm for 3-4 minutes.
Serve with grated parmesan and fresh basil, optionally accompanied by a green vegetable and garlic bread.