1. HEAT OVEN TO 375F. GREASE AND FLOUR 15×10-INCH JELLY ROLL PAN. FOR THE BARS, IN A LARGE BOWL, BEAT TOGETHER SUGAR, SOUR CREAM, BUTTER, AND EGGS UNTIL CREAMY. BLEND IN BANANAS AND VANILLA EXTRACT. ADD FLOUR, BAKING SODA, SALT, AND BLEND FOR 1 MINUTE. STIR IN WALNUTS.
2. SPREAD BATTER EVENLY INTO PAN. BAKE 20 TO 25 MINUTES OR UNTIL GOLDEN BROWN.
3. MEANWHILE, FOR FROSTING, HEAT BUTTER IN A LARGE SAUCEPAN OVER MEDIUM HEAT UNTIL BOILING. LET THE BUTTER TURN A DELICATE BROWN AND REMOVE FROM HEAT IMMEDIATELY.
4. ADD POWDERED SUGAR, VANILLA EXTRACT, AND MILK. WHISK TOGETHER UNTIL SMOOTH (IT SHOULD BE THICKER THAN A GLAZE BUT THINNER THAN FROSTING). USING A SPATULA, SPREAD THE BROWN BUTTER FROSTING OVER THE WARM BARS (THE FROSTING WILL BE EASIER TO SPREAD WHILE THE BARS ARE STILL WARM).