1. In a 3- to 4-qt. pan over high heat, bring 3 cups water to a boil. Gradually stir in polenta. Reduce heat and simmer gently, uncovered, stirring frequently to prevent sticking, until mixture is very thick, about 25 minutes. Stir in 2 tbsp. butter.
2. Spoon polenta into a buttered 8- to 9-in. square pan; spread in an even layer and let stand at least 10 minutes (or cool, cover, and refrigerate up to overnight). Cut polenta into 4 squares and gently remove from pan with a wide spatula.
3. Melt 2 tbsp. butter in a large frying pan over medium-high heat. Add polenta squares and cook, turning once, until crispy and lightly browned on both sides, about 10 minutes. Add more butter, if needed, to prevent sticking. Lift out to a warm platter; keep warm. Add prosciutto to pan and cook until meat is lightly browned (add more butter to pan, if needed). Remove and keep warm.
4. Fill a large, clean frying pan with about 1 1/2 in. of water. Set over medium-high heat, add vinegar, and heat until bubbles form on pan bottom, with an occasional bubble popping to the top. Break eggs into water, keeping them slightly apart. Cook until egg whites are firm but yolks are still soft, 3 to 4 minutes. Gently remove eggs with a slotted spoon.
5. Place 2 or 3 slices prosciutto on each polenta square, top each with a poached egg, then spoon pesto sauce evenly over top.