Line an 8- or 9-inch square baking pan with parchment paper, leaving 2 to 3 inches overhang on the sides.
Combine milk chocolate chips, semisweet chocolate chips, condensed milk, and salt in a large microwave-safe bowl. Microwave on high until mostly melted, about 2 minutes. Stir with a soft spatula until creamy. Mix in peanut butter and microwave for 1 minute. Add vanilla extract and stir until smooth. Transfer to the prepared baking pan and smooth top with a spatula.
Combine peanut butter chips and milk in a small microwave-safe bowl. Microwave on high for 1 minute. Stir, and continue to microwave in 30 second increments, stirring in between, until mixture is smooth. Pour on top of the fudge mixture and smooth top with a spatula. Place in the refrigerator until firm, about 2 hours.