Preheat the broiler. Put the poblano cut-side down on a baking sheet and broil until the skin is browned in spots, about 5 minutes. Transfer to a bowl, cover with plastic wrap and let cool 5 minutes. Peel the skin off the poblano under cold running water, then chop into small pieces.
Preheat the oven to 400 degrees F. Lightly butter a 2-quart baking dish. Heat 2 tablespoons butter in a large skillet over medium-high heat. Add the onion, garlic and thyme and cook, stirring frequently, until the onion is soft, about 5 minutes. Add the potatoes, chicken broth, 1 1/2 teaspoons salt, and pepper to taste and bring to a boil. Reduce the heat to medium and cook, stirring gently, until the potatoes are tender, 10 to 12 minutes.
Arrange half of the potato-broth mixture in the prepared baking dish in an even layer. Sprinkle with half each of the roasted poblano and parmesan. Repeat with the remaining potato-broth mixture, poblano and parmesan. Bake until bubbly and slightly golden, about 35 minutes. Let stand 15 minutes before serving. Thin some crema or sour cream with water and serve alongside for topping.