Turn your oven broiler on high and line a baking sheet with foil. Place the whole hatch chilies on the sheet and slide them a few inches under the broiler. Let them blacken and blister for about 10 to 15 minutes, turning every few minutes so all sides get charred evenly.
Once the skins are blackened, transfer the hot chilies straight into a zip-top bag or a bowl covered tightly with plastic wrap. Let them sit for about 10 minutes so the trapped steam loosens the skins. After that, the charred skin should peel off with almost no effort at all.
Remove the stems and seeds from the peeled chilies, then roughly chop them. I like to puree about half of the chopped chilies in a blender until smooth, and leave the rest chunky. This combination gives the chili a rich sauce with plenty of texture in every spoonful.
Heat olive oil in a large Dutch oven over medium-high heat. Add the diced onion and cook for about 4 minutes until it turns soft and translucent. Stir in the minced garlic and cook for another minute, just until it becomes fragrant.
Add the cubed chicken thighs to the pot and season with cumin, smoked paprika, salt, and pepper. Cook for 5 to 7 minutes, stirring occasionally, until the outside of the chicken turns golden brown. This step locks in flavor before the liquid goes in.
Pour in the pureed and chopped hatch chilies along with the chicken broth. Stir everything together, then bring the pot to a gentle boil. Reduce the heat to low, cover, and let it simmer for 45 minutes so the flavors have time to deepen.
Once the chicken is tender and the broth has thickened slightly, stir in the fresh lime juice and taste for seasoning. Ladle the chili into bowls and top with chopped cilantro right before serving for a bright, fresh finish.