In a medium saucepan, combine the white rice, chicken broth, and water. Bring to a boil over medium-high heat. Stir once, reduce the heat to low, cover tightly, and simmer for 15–18 minutes, or until the liquid is absorbed and the rice is tender. Remove from the heat and let stand, covered, for 5 minutes. If your rice variety requires additional liquid, follow the package's recommended total amount.
While the rice cooks, drain and finely chop the roasted red pepper. Chop the scallions and set both aside.
In a small bowl, whisk together the Dijon mustard, honey, apple cider vinegar, olive oil, tarragon, and minced garlic until smooth. Transfer 2 tablespoons of the sauce to a separate clean bowl for serving.
Pat the chicken breasts dry with paper towels. If necessary, gently pound the thicker portions to approximately ¾-inch thickness for even cooking. Brush both sides with olive oil and season with salt and pepper.
Heat a large grill pan over medium heat for 2–3 minutes, until hot. The pan should be hot enough to create grill marks without scorching the honey mustard sauce.
Place the chicken breasts in the grill pan in a single layer, working in batches if necessary. Cook for 5–6 minutes on the first side, until golden brown with visible grill marks.
Turn the chicken and brush the cooked side with the honey mustard sauce. Continue cooking for another 5–7 minutes, turning once more if necessary and brushing with the remaining grilling sauce during the final few minutes. Reduce the heat slightly if the glaze begins to brown too quickly. Cook until the thickest part of each chicken breast reaches an internal temperature of 165°F.
Transfer the chicken to a clean platter and let rest for 5 minutes.
Fluff the cooked rice with a fork. Gently fold in the chopped roasted red peppers and scallions until evenly distributed. Cover briefly to warm the vegetables through.
Divide the roasted red pepper rice evenly among four plates. Add one chicken breast to each plate, drizzle with the reserved honey mustard sauce, and serve immediately.