Poach the chicken: In a large pot set over medium-high heat, add the olive oil, tomatoes, onion, carrot and salt. Reduce the heat to medium and cook, stirring occasionally, until the mixture turns thick and pasty, about 10 minutes (don't let it brown). Add the water, cilantro stems and chile. Add the chicken breasts, rounded side up—they should just be barely covered with water. Reduce the heat to medium-low so the liquid is steaming but not bubbling (reduce the heat even more if necessary). Cook for 45 minutes and then turn off the heat (the chicken can be refrigerated overnight in the broth for a more full-flavored dish, or it can be strained right away).
Remove the chicken from the broth and set aside to cool, then shred the chicken. Strain the broth through a fine-mesh sieve and discard the solids.
Make the rice: Add the olive oil to a large skillet set over medium-high heat. Once it shimmers, add the sage and thyme and let it sizzle for 10 seconds. Add the onion and salt and cook, stirring occasionally, until the onion is soft, about 4 minutes. Stir in the rice and add 4 cups of the broth (save the rest for another time). Bring to a simmer, reduce the heat to low, cover the skillet and cook until the rice is tender, 50 to 55 minutes.
Stir in the chicken and turn off the heat. Place a paper towel over the skillet and cover. Set aside for 10 minutes. Remove the paper towel and serve the rice sprinkled with the cilantro.