Place chicken in a skillet or saucepan and add a bit of water; Add lemon juice and bring to a simmer over high heat. Cover, reduce heat and simmer gently until the chicken is cooked through and no longer pink in the middle, 10 to 12 minutes. Transfer to a cutting board. When cool enough to handle, shred with two forks into bite-size pieces. Add salt, pepper, and chili powder to chicken and combine.
Bring a large pot of water to a boil, add broccoli and cook until tender, 3 to 5 minutes. Drain and rinse with cold water until cool. Meanwhile, core tomatoes and cut in half crosswise. Gently squeeze out seeds and discard. Set the tomatoes cut-side down on paper towels to drain for about 5 minutes.
Place a large heavy skillet, such as cast-iron, over high heat until very hot. Add the oil to the skillet along with the tomatoes and place cut-side down in the pan. Cook until charred and beginning to soften, 4 to 5 minutes. Turn and cook until the skin is charred, 1 to 2 minutes more. Transfer to a plate to cool. Do not clean the pan.
Combine tomatoes in a large bowl with the shredded chicken, boiled broccoli.
