Preheat oven to 375°F (190°C) and grease a 10x15-inch baking sheet. In a large bowl, beat granulated sugar, sour cream, butter, and eggs until smooth and creamy.
Fold in mashed bananas and vanilla extract until combined.
Stir in flour, baking soda, and salt, being careful not to overmix. Spread the batter evenly in the baking sheet.
Melt butter in a saucepan over medium heat, cooking until golden brown and nutty (about 5-7 minutes). Remove from heat and whisk in powdered sugar, vanilla, and milk until smooth. Add more milk if needed for consistency.
Pour the glaze over the warm bars, spreading evenly. Let it set before cutting into squares.