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Special equipment: 2-inch round cutter
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For the cupcakes: Preheat the oven to 350 degrees F and position a rack in the middle of the oven.
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Line one 12-cup standard muffin tin with cupcake liners and spray generously with baking spray.
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Whisk together the flour, cornmeal, sugar, baking powder, 1 teaspoon salt and the pepper in a large bowl.
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Set aside.
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Heat 2 tablespoons softened butter in a medium skillet over medium-high heat.
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Once the butter begins to foam, add the onions and thyme and cook, stirring frequently, until soft and dark brown, about 8 minutes.
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Season with the remaining 1/2 teaspoon salt.
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Remove from the heat and stir in the chicken and grated carrot, just to warm through.
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Set aside to cool.
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Whisk together the milk, sour cream, eggs and 2 tablespoons melted butter.
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Toss the chicken-onion mixture into the flour and then stir in the milk-egg mixture until just combined (take care not to over mix).
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Evenly divide among the prepared muffin cups and bake until a toothpick inserted in the center comes out clean, 20 to 22 minutes.
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Cool the cupcakes on a rack in the tin for 10 minutes, and then remove.
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Cool on the rack completely.
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For the crust: Increase the oven to 425 degrees F and line a baking sheet with parchment paper.
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Cut the puff pastry into twelve 2-inch rounds and arrange on the prepared baking sheet.
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Poke each round a few times with a fork, cover with another sheet of parchment and weigh down with another baking sheet.
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Bake for 10 minutes.
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Remove the top baking sheet and parchment paper and sprinkle with the thyme leaves, salt and pepper.
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Return to the oven and bake until golden brown and crispy, about 5 minutes more.
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Allow to cool (these can be made a day ahead).
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For the sweet pea cream: Combine the peas and 1 tablespoon water in a small microwave-safe bowl.
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Cover with plastic wrap and cook for 1 minute.
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Combine the cream cheese, sour cream, lemon zest, salt and pepper to taste in the bowl of a food processor.
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Pulse until blended and smooth.
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Add the peas and parsley, and pulse once or twice just to roughly chop the peas.
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Scrape into a small bowl, cover with plastic wrap and refrigerate until ready for use (this can be made a day ahead).
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To assemble: Top the cupcakes with a heaping tablespoon of sweet pea cream and spread to coat.
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Decorate each with a piece of chicken, chopped carrots and peas.
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Top with a crust.
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Serve immediately.
