Cut chicken cross-wise into 1/4?-wide slices. Put the chicken back into the skillet and coat slices with the juices remaining in the pan.

In a medium bowl, add 3 tablespoons vegetable oil, lime juice, pressed garlic, paprika, sugar, salt, cumin, pepper, and cayenne together. Whisk to combine, add the chicken to the bowl and toss to evenly coat. Cover bowl with plastic wrap and allow to marinade at room temperature for 1 hour.

Meanwhile, set an oven rack as close as possible to the broiler element. Slice poblanos chiles in half, remove the stem and seeds, and arrange with the skin-side up onto a baking sheet lined with foil. Some of the chiles may need to be flattened so that they are the same distance from the broiler.

Broil for 7 to 10 minutes until the skin becomes charred, rotating and adjusting the tray so that they char evenly.
Empty the chiles into a large bowl, cover tightly with plastic wrap, and allow them to steam for 10 minutes. Rearrange the oven racks so that there is one in the middle and lower positions and set oven to 200?F.

Empty the chiles into a large bowl, cover tightly with plastic wrap, and allow them to steam for 10 minutes. Rearrange the oven racks so that there is one in the middle and lower positions and set oven to 200?F.

Peel away most of the skin from the chiles, and slice into 1/4? wide strips. Remove onions ends, slice in half, then slice onion from pole-to-pole into 1/4? wide strips.

Add 2 tablespoons vegetable oil to a 12? non-stick skillet. Preheat over a high burner until the oil just begins to smoke. Add onions and cook for 4 minutes until they become charred. Cook the pressed garlic, thyme and dried oregano for just 15 seconds. Add the 1/2 cup of cream and reduce for 1-1/2 to 2 minutes until the onions are lightly coated with cream. Add sliced chile, lime juice, salt and pepper to the skillet, mix until evenly coated with cream.

Empty the veggies into the same bowl you used to steam the chiles. Cover with aluminum foil and set on middle rack in the oven.

Use paper towels to wipe out the skillet. Remove the chicken from the bowl and use more paper towels to wipe of some excess, but not all of the marinade

Set skillet over high burner and preheat 1 tablespoon vegetable oil until it begins to smoke. Cook chicken for 4 minutes without moving; until the bottom becomes charred. Flip chicken, cook for 2 minutes until the chicken registers 150?F. Bake for about 10 minutes until chicken internal temperature of the chicken reaches 160?F

Put cooked chicken on cutting board, tent loosely and allow to rest for 5 minutes. Don?t wash the skillet.

When serving, put a few chicken slices in a warm flour tortilla. Put some vegetable mixture, cilantro ontop. Squeeze a lime wedge.