In a large pot, bring 3 cups water and 3 cups whole milk to a low boil.
Whisk in 2 cups polenta, reduce heat to medium-low, and cook, stirring often, 15–20 minutes until thickened.
Stir in 2 tablespoons butter and 3/4 cup shredded sharp cheddar until melted, then fold in the corn kernels and season with kosher salt and pepper to taste.
Preheat a grill or grill pan to medium-high heat.
In a large zip-top bag or bowl, combine 1 1/2 pounds chicken and 1 cup BBQ sauce with 1/4 cup honey; toss to coat evenly.
Grill the chicken 3–4 minutes per side, flipping gently 2–3 times, brushing with any remaining sauce, until cooked through and lightly charred.
Remove the chicken and let rest 5 minutes, then slice against the grain.
Divide the polenta among bowls, top with sliced peaches, sliced avocado, and the sliced BBQ chicken.
Garnish with fresh cilantro and chopped chives and serve with extra BBQ sauce if desired.