For the cupcakes: Preheat the oven to 350 degrees F and position a rack in the middle of the oven. Line one 12-cup standard muffin tin with cupcake liners and spray generously with baking spray.
Whisk together the flour, cornmeal, sugar, baking powder, 1 teaspoon salt and the pepper in a large bowl. Set aside.
Heat 2 tablespoons softened butter in a medium skillet over medium-high heat. Once the butter begins to foam, add the onions and thyme and cook, stirring frequently, until soft and dark brown, about 8 minutes. Season with the remaining 1/2 teaspoon salt. Remove from the heat and stir in the chicken and grated carrot, just to warm through. Set aside to cool.
Whisk together the milk, sour cream, eggs and 2 tablespoons melted butter. 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). Evenly divide among the prepared muffin cups and bake until a toothpick inserted in the center comes out clean, 20 to 22 minutes. Cool the cupcakes on a rack in the tin for 10 minutes, and then remove. Cool on the rack completely.
For the crust: Increase the oven to 425 degrees F and line a baking sheet with parchment paper. Cut the puff pastry into twelve 2-inch rounds and arrange on the prepared baking sheet. Poke each round a few times with a fork, cover with another sheet of parchment and weigh down with another baking sheet. Bake for 10 minutes. Remove the top baking sheet and parchment paper and sprinkle with the thyme leaves, salt and pepper. Return to the oven and bake until golden brown and crispy, about 5 minutes more. Allow to cool (these can be made a day ahead).
For the sweet pea cream: Combine the peas and 1 tablespoon water in a small microwave-safe bowl. Cover with plastic wrap and cook for 1 minute. Combine the cream cheese, sour cream, lemon zest, salt and pepper to taste in the bowl of a food processor. Pulse until blended and smooth. Add the peas and parsley, and pulse once or twice just to roughly chop the peas. Scrape into a small bowl, cover with plastic wrap and refrigerate until ready for use (this can be made a day ahead).
To assemble: Top the cupcakes with a heaping tablespoon of sweet pea cream and spread to coat. Decorate each with a piece of chicken, chopped carrots and peas. Top with a crust. Serve immediately.