Combine chocolate chips and 3 Tbsp. water in a microwave-safe bowl and microwave on high until just melted, 1 to 1 1/2 minutes. Whisk until smooth.
Place bananas in a food processor and pulse until broken down. Add milk and vanilla and process until smooth, scraping down sides of bowl once or twice as necessary.
Scoop banana puree into dishes and top with warm chocolate sauce. Serve immediately.
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