Preheat oven to 375°F. Grease a 9x13-inch pan or line with parchment paper.
Cream sugar and softened butter until fluffy. Add Greek yogurt, eggs, mashed bananas, and vanilla; mix until smooth. Stir in flour, baking soda, and salt until just combined—don’t overmix.
Pour batter into pan and smooth the top. Bake for 23–28 minutes, until golden on the edges and a toothpick comes out clean.
Melt butter in a saucepan over medium heat until golden brown and fragrant.
Remove from heat and whisk in powdered sugar, vanilla, and 1 tablespoon milk. Add more milk as needed for a pourable but thick consistency.
While bars are still warm, pour frosting over the top. Spread evenly with a spatula—let it soak in slightly for a glossy finish.