Warm the milk, sprinkle yeast and a pinch of sugar, and let sit 5–10 minutes until foamy. Mix melted butter, sugar, and egg in a bowl. Add yeast mixture, then flour and salt. Knead until smooth and elastic (6–8 minutes by hand or 5–6 minutes with a mixer). Let rise in an oiled bowl until doubled, about 1–1.5 hours.

Combine diced peaches, cornstarch, brown sugar, lemon juice, and peach jam in a small saucepan. Cook over medium heat, stirring until thickened and peaches soften (3–5 minutes). Cool.

Punch down the dough and roll to a 12×18-inch rectangle. Spread softened butter, sprinkle cinnamon sugar, then spoon cooled peach filling evenly, leaving a 1/2-inch border. Roll tightly and slice into 9–12 rolls. Place in a greased 9×13-inch pan, cover, and let rise 30–45 minutes.

Preheat oven to 350°F (175°C). Bake rolls 25–30 minutes until golden and set. Whisk powdered sugar, milk, and vanilla until smooth, then drizzle over warm rolls.