Place chicken in a skillet or saucepan and add enough water to cover; 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.

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. Brush the cut sides of the tomatoes with 1 teaspoon oil and place cut-side down in the pan. Cook until charred and beginning to soften, 4 to 5 minutes. Brush the tops lightly with another 1 teaspoon oil, turn and cook until the skin is charred, 1 to 2 minutes more. Transfer to a plate to cool. Do not clean the pan.

Heat the remaining 3 tablespoons oil in the pan over medium heat. Stir in salt, pepper and chili powder and cook, stirring constantly, until fragrant, about 45 seconds. Slowly pour in lemon juice (it may splatter), then remove the pan from the heat. Stir to scrape up any browned bits.

Coarsely chop the tomatoes and combine them in a large bowl with the shredded chicken, broccoli and the pan dressing; toss to coat.

MAKE AHEAD TIP: Cover and refrigerate for up to 2 days. Add more lemon and salt, if needed, just before serving.