For polenta, bring stock and milk to the boil in a saucepan over high heat. Add polenta in a thin, steady stream, whisking continuously until incorporated, then reduce heat to medium-low and cook, whisking occasionally, until thick and no longer grainy (20-25 minutes). Cover with a round of baking paper and keep warm. Just before serving, whisk in parmesan and butter and season to taste.
Preheat oven to 220C. Melt butter in a saucepan over high heat, then cook until it turns nut-brown (2-4 minutes). Add garlic and herbs, transfer to a bowl and set aside.
Wash quail and pat dry with absorbent paper, wrap each in 1 wide or 2 thin prosciutto slices, then transfer to a small roasting pan. Brush quail with herb butter, season with freshly ground black pepper and roast until golden and just cooked through (20-25 minutes). Set aside to rest for 5 minutes.
Divide polenta among warmed plates, top with quail, spoon herb butter on top and serve with lemon wedges on the side.