Mix the cornstarch with ¼ cup chicken stock and set aside.
In a large skillet or Dutch oven, warm the olive oil over medium-high heat and then add the chopped carrots. Season with salt and pepper and cook until the carrots just start to caramelize, about 5 minutes. Add 1 ¾ cups chicken stock, the cream or coconut milk, chopped chicken, frozen onions and minced rosemary and bring to a simmer. You can also add any roasted veggies you may have on hand, butternut squash or potatoes are particularly good. Let simmer about 15 minutes. The veggies should be tender and the chicken heated through. Add the frozen peas. Turn the heat up; bring the mixture to a boil and stir in the cornstarch mixture. Boil for 1 minute and take off the heat. Taste the mixture and adjust seasoning if needed. Divide the mixture between 4 oven proof bowls or ramekins and set them on a baking sheet.
Preheat the broiler and place the oven rack in the middle.
In a saucepan over medium high heat, bring the remaining cup of chicken stock and remaining half cup of cream or coconut milk to a boil. Gradually whisk in the polenta and cook until thick, about 3 minutes. Turn the heat to low and stir in the cheese until melted.
Top the chicken mixture with the polenta and smooth it. Place in pre-heated broiler for 2 -3 minutes or until the tops are browned. Serve with a garnish of rosemary sprigs if desired.