Adjust oven rack to highest position and preheat broiler.
Whisk together 2 T oil, 4 t lime juice, garlic, paprika, sugar, salt, cumin, black pepper and cayenne pepper. Add chicken and toss to coat. Cover and let stand at room temperature 30-60 minutes.
Line a rimmed baking sheet with aluminum foil and place poblanos on it, skin side up, then press them flat. Broil until skin is charred and puffed, 4-10 minutes, rotating baking sheet 180 degrees every 2 minutes. Transfer poblanos to a bowl and cover tightly with plastic wrap; let steam for 10 minutes. Rub off most of the skin from the peppers (some skin/char may be left for flavor). Slice peppers into 1/4" strips.
Adjust oven racks to lowest and middle positions and turn oven on at 200°F.
Heat 2 t oil in 12" nonstick skillet over high heat until wisps of smoke appear. Add onion and cook until charred and slightly soft, 3 minutes. Add garlic, thyme, and oregano, and cook until fragrant, 15-30 seconds. Add cream and stir often until cream thickens and clings to onions, 1-2 minutes. Add poblano strips, 2 t lime juice, salt, and pepper, and toss to coat. Transfer vegetables to oven-safe bowl and place on middle oven rack to keep warm. Wipe out pan with paper towel.
Add remaining 1 T oil and heat over high heat until smoking. Remove chicken from marinade and wipe off excess. Add chicken and cook without moving until bottom side is well charred, about 4 minutes. Flip chicken and transfer skillet to lower oven rack. Bake until internal temperature of chicken reaches 160F, 7-10 minutes. Transfer to cutting board and let rest 5 minutes. Do not wash out skillet.
Slice cooled chicken into 1/4" thick strips. Return chicken strips to pan and toss to coat with pan juices. Transfer to a bowl to serve.
To serve, place a few pieces of chicken into the middle of a warmed tortilla, top with a spoonful of vegetable mixture, cilantro and cheese.