Grease a 9×13-inch pan with butter and warm it in a draft-free space.
In a saucepan, combine milk, 3 tbsp sugar, and ⅓ cup butter. Heat until melted, then cool to 100-115°F.
Add instant yeast and ¾ cup warm water to the milk mixture in a mixer bowl. Mix on low for 1 minute.
Gradually add 2 cups flour and ½ tsp salt on low speed until dough cleans bowl edges.
Knead dough for 2 more minutes until elastic. Let rise 10 minutes in a warm area.
Whisk brown sugar, ¼ cup sugar, corn syrup, ¼ tsp salt, and 2 tbsp water for filling. Spread on rolled-out dough.
Roll dough into a 12×16-inch rectangle, spread filling, then roll lengthwise. Cut into 12 rolls. Place cut-side down in the pan.
Bake at 375°F (190°C) for 35–40 minutes until golden and bubbly.