Make the polenta: Bring the broth to a boil in a saucepan over medium-high heat. Gradually whisk in the polenta and cook, whisking, until thick and creamy. Season with salt and pepper and stir in the sage and pineapple. Brush a 9-inch square baking dish with vegetable oil; add the polenta and spread evenly. Press a piece of plastic wrap on the surface of the polenta and refrigerate until firm, at least 1 hour.

Prepare the shrimp: Puree the pineapple in a blender until smooth. Place 1/4 cup of the puree in a bowl (reserve the rest for another use) with the shrimp, olive oil, garlic, red pepper flakes, and salt and pepper to taste. Toss, then cover and marinate at room temperature, about 20 minutes. Remove the shrimp from the marinade and wrap a strip of prosciutto around each piece.

Make the sauce: Melt the butter in a small skillet over medium-low heat and cook until brown flecks appear. Remove from the heat and stir in the pineapple.

Preheat a grill or grill pan to medium and brush with vegetable oil. Cut the polenta into 12 squares and grill until marked, 2 to 3 minutes per side. Grill the prosciutto-wrapped shrimp until the shrimp is pink, 1 to 2 minutes per side.

Serve the shrimp on polenta squares, drizzle with the butter sauce and garnish with parsley.

Photograph by Stephanie Foley