Line an 8x8 or 9x9 square pan with foil (or parchment) and spray with nonstick cooking spray.
Important: when you're making fudge it's important to have everything ready to go before you begin so you can work quickly once the chocolate melts. Also - if you wash the pan in between layers (instead of using a second pan) be sure it is completely dry. Water kills chocolate!
Place the milk or semi-sweet chocolate chips and 1/2 cup sweetened condensed milk in a medium saucepan over medium-low heat. Cook, stirring constantly, until melted and smooth. Remove from heat and stir in a pinch of salt, 1/2 teaspoon vanilla, and Oreo pieces. Spread in prepared pan. If it's cool in your house, let it sit for 15-20 minutes, otherwise chill it to set the layer slightly. You don't want it firm, just not smooshy.
Place the white chocolate chips and remaining 1/2 cup sweetened condensed milk in a second medium saucepan (or the washed and dried one) over medium-low heat. Cook, stirring constantly, until melted and smooth. Remove from heat and stir in a pinch of salt, 1/2 teaspoon vanilla, and peppermint/candy cane pieces. Spread over chocolate layer. Sprinkle with remaining crushed candy canes.
Let cool to room temperature and then chill to set. (My house was cool enough at 60-65° overnight to set the fudge without having to chill it.) Slice into squares and serve or package for giving. Store in an airtight container for up to one week.