Add ⅔ cup granulated sugar, 2 tablespoons cornstarch, and ⅛ teaspoon salt to small saucepan. Whisk dry ingredients together to blend.
Place saucepan on stovetop over medium-high heat. Immediately pour 2 cups whole milk into saucepan in increments, whisking well to incorporate milk between each addition.
Once all milk has been added, continue whisking gently until mixture is smooth, bubbly, and thickened. Once bubbly and thickened, reduce heat to low. Gently stir mixture over low heat 2 minutes.
After 2 minutes, remove saucepan from heat. Transfer 2 tablespoons of milk mixture into bowl of 4 large egg yolks (lightly beaten) and stir gently to incorporate. Add milk mixture slowly to prevent scrambling egg yolks.
Slowly pour egg mixture into saucepan, stirring constantly. When entire egg mixture has been transferred to saucepan, return saucepan to heat.
Increase heat under saucepan to medium. Stirring constantly, bring mixture to very low, gentle boil. Cook mixture, stirring constantly, 2 minutes.
After 2 minutes, remove saucepan from heat. Let mixture cool slightly, then add 1 ½ teaspoons pure vanilla extract and stir to incorporate.
Transfer mixture to glass or ceramic bowl. Let mixture cool 10 to 15 minutes.
Once mixture has cooled, cover bowl with food-safe plastic wrap, lightly pressing plastic wrap against top of custard. Place covered custard in refrigerator and chill at least 1 hour.
Immediately before serving, peel 2 medium ripe bananas, then chop bananas into small pieces.
Remove chilled custard from refrigerator and discard plastic wrap. Add chopped banana pieces to bowl of custard, then gently fold banana pieces into custard until just incorporated. Serve banana custard chilled or at room temperature as desired.