Season the chicken breasts with 2 of the garlic cloves, salt, and pepper, and drizzle with about ½ tablespoon of the oil. Grill the chicken breasts or bake them in a roasting pan, uncovered at 350°F for 20 minutes or until cooked. Alternatively, mark the chicken breasts on a grill or grill pan (to get grill marks) and then finish them in the oven until they are cooked through.
Meanwhile, take the poblano chile and either grill it, broil it, or hold it with tongs over a gas flame to char the skin. When the skin has charred, remove it from the heat source and place it in a plastic bag for a few minutes to steam (this will make it easier to peel). Peel the poblano, remove the seeds, and slice into thin strips.
Heat the remaining 2 tablespoons oil in a large heavy skillet over medium-high heat. Add the peppers, onion, chorizo, and the remaining minced garlic clove. Cook, stirring occasionally until the onions and peppers are soft. Add the sour cream, half-and-half, and season with salt and pepper. Simmer for 3 minutes until the sauce starts to thicken.
Spoon the vegetable mixture evenly onto four serving dishes and top each with a chicken breast. Sprinkle crumbled queso fresca over the top, if desired.