Preheat oven to 350 degrees F (175 degrees C). Grease 12 muffin cups or line with paper liners.
Mix peaches, basil, and 1 tablespoon brown sugar in a bowl; let sit until sugar dissolves, about 15 minutes.
Sift flour, baking powder, baking soda, and salt together in a large bowl. Mix 1 cup Cheddar cheese into flour mixture.
Beat butter and 7 tablespoons brown sugar in a bowl with an electric mixer until smooth and creamy. Add eggs, one at a time, until just combined; gently fold in peaches. Mix flour mixture, a little at time, into peach mixture until batter is just combined.
Spoon batter into the prepared muffin cups. Sprinkle 1/4 cup Cheddar cheese over batter.
Bake in the preheated oven until a toothpick inserted in the middle of a muffin comes out clean, about 25 minutes.
