Heat grill on medium-high. Halve chicken breasts laterally, making sure not to go all the way through (each should open like a book). Using meat mallet, pound center seam to even thickness. Season with ½ teaspoon each salt and pepper.
In large bowl, toss broccoli with 1 tablespoon oil and ½ teaspoon salt. Grill chicken until charred and just cooked through, about 2 minutes per side. Grill broccoli, turning 2 to 3 times, until charred and just tender, 3 to 4 minutes. Transfer to cutting board, roughly chop, then return to bowl.
In jar, combine cherry juice, oil, vinegar, mustard, and ⅛ teaspoon each salt and pepper.
Pour dressing over broccoli and toss to combine, then toss with cashews, dried cherries, and mint leaves. Fold in 2 tablespoons Pecorino Romano. Serve over chicken and sprinkle with remaining 2 tablespoons Pecorino Romano.
