Brush halved peaches with melted butter. Place on a heated grill, or grill pan, over medium-high heat. Grill until just soft and warm throughout (approximately 3-4 minutes, turning as needed).
Meanwhile, or once the peaches have finished cooking, whisk together brown sugar and honey in a small saucepan. Heat on low and simmer for 2 minutes, until sugars have dissolved. Set aside.*
Top each grilled peaches with approximately 1 teaspoon of mascarpone, a drizzle of honey sauce, and a few basil leaves.
Notes
The honey sauce will harden as it cools, so time it out that you can use it shortly after it's dissolved together.
Adapt to the amount of peaches you want (or need)! You'll likely have extra ingredients to continue enjoying!