Adjust oven racks to middle and highest positions and heat broiler. Heat 2 teaspoons oil in medium saucepan over medium heat until shimmering; add onion and cook, stirring frequently, until golden, 6 to 8 minutes. Add 2 teaspoons garlic and cumin; cook, stirring frequently, until fragrant, about 30 seconds. Decrease heat to low and stir in broth. Add chicken, cover, and simmer until instant-read thermometer inserted into thickest part of chicken registers 160 degrees, 15 to 20 minutes, flipping chicken halfway through cooking. Transfer chicken to large bowl; place in refrigerator to cool, about 20 minutes. Remove 1/4 cup liquid from saucepan and set aside; discard remaining liquid.
Meanwhile, toss tomatillos and poblanos with remaining 2 teaspoons oil; arrange on rimmed baking sheet lined with foil, with poblanos skin-side up. Broil until vegetables blacken and start to soften, 5 to 10 minutes, rotating pan halfway through cooking. Cool 10 minutes, then remove skin from poblanos (leave tomatillo skins intact). Transfer tomatillos and chiles to food processor. Decrease oven temperature to 350 degrees. Discard foil from baking sheet and set baking sheet aside for warming tortillas.
Add 1 teaspoon sugar, 1 teaspoon salt, remaining teaspoon garlic, and reserved cooking liquid to food processor; process until sauce is somewhat chunky, about eight 1-second pulses. Taste sauce; season with salt and pepper and adjust tartness by stirring in remaining sugar, 1/2 teaspoon at a time. Set sauce aside (you should have about 3 cups).
When chicken is cool, pull into shreds using hands or 2 forks, then chop into small bite-sized pieces. Combine chicken with cilantro and 11/2 cups cheese; season with salt.
Smear bottom of 13- by 9-inch baking dish with 3/4 cup tomatillo sauce. Place tortillas on 2 baking sheets. Spray both sides of tortillas lightly with cooking spray. Bake until tortillas are soft and pliable, 2 to 4 minutes. Increase oven temperature to 450 degrees. Place warm tortillas on countertop and spread 1/3 cup filling down center of each tortilla. Roll each tortilla tightly and place in baking dish, seam-side down. Pour remaining tomatillo sauce over top of enchiladas. Use back of spoon to spread sauce so that it coats top of each tortilla. Sprinkle with remaining 1/2 cup cheese and cover baking dish with foil.
Bake enchiladas on middle rack until heated through and cheese is melted, 15 to 20 minutes. Uncover, sprinkle with scallions, and serve immediately.