Slice peaches in half lengthwise and remove pits; set aside.
In a small bowl, mix butter and cinnamon sugar until well combined.
Heat a grill pan (or outdoor grill) to medium-high. Brush peaches with oil and grill cut-side-down until slightly softened and charred, a little over 1 minute. Compress gently with a spatula a few times for grill marks. Flip to rounded side and grill an additional minute.
Serve grilled peaches on individual plates. Top each with cinnamon-sugar butter, fresh basil or mint, and a scoop of vanilla ice cream or whipped cream.
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