Cost per Serving:
Lightly grease a large, rimmed baking sheet. In a bowl, blend cornmeal and 1 cup cold broth into a thick paste.
In a large saucepan, bring remaining broth and butter to a rolling boil. Whisk in puree, lower heat and return to a gentle boil. Whisk in cornmeal and cook, stirring constantly, until thick enough to hold trail of a spoon, about 5 minutes.
Remove polenta from heat, stir in cheese and thyme; season with salt and pepper. Stir until cheese is melted and mixture is blended. Working quickly, pour hot polenta into prepared pan and spread evenly with a spatula. Cool for 1 hour, cover and refrigerate for at least 2 hours, until chilled through and firm. (Polenta can be made to this point up to 2 days ahead.)
Heat oven to 400°F. Lightly grease 2 nonstick baking sheets. Brush top of polenta with olive oil. Cut out 2- to 2 1/2-inch shapes with a pumpkin cookie cutter (or other shape). Use a spatula to transfer polenta to baking sheets. Bake until lightly browned, about 15 minutes, rotating pans halfway through. Serve warm, topped with a scallion piece.