Heat the oil in a large Dutch oven or pot over medium-high until shimmering. Add the onion and carrots, season lightly with salt and cook, stirring occasionally, until the onions just start to soften, about 3 minutes.
Push the carrots and onions to the sides of the pot, creating space in the center. Season the chicken thighs all over with salt and pepper, then add them to the center of the pot. Cook until the skin easily releases from the pan, 7 to 9 minutes, pushing the onions and carrots occasionally.
Add the tomato paste, cumin, turmeric and garlic to the carrots and onions, stirring as best you can. Flip the chicken, stack the pieces to make some room in the pot to stir, and cook until the tomato paste intensifies and darkens in color, about 2 minutes.
Add the lentils and 4 cups of water/broth (or more if needed, to fully submerge the lentils and most of the chicken). Adjust the heat to bring the liquid to a boil, and season with salt. Cover with the lid slightly ajar, adjust the heat to maintain a simmer, and cook, stirring occasionally, until the lentils are tender, the chicken is cooked through and the flavors blended, 40 to 45 minutes.
Stir in half the lime juice, spoon some sauce over the chicken, then taste and season as needed with more lime juice or salt. Finish with a few grinds of pepper and sprinkle with the cilantro before dividing among bowls.
Store in refrigerator for up to 3 days.