Preheat and bake the frozen deep-dish pie crust according to package directions, pricking the bottom and sides with a fork to prevent air pockets; set the baked crust aside to cool.
In a saucepan over medium-high heat, combine the granulated sugar and eggs and stir constantly until the mixture reaches 160°F (about the temperature at which it thickens); immediately transfer to a large bowl to stop cooking.
Melt the semi-sweet baker's chocolate in a microwave-safe bowl in 30-second intervals, stirring after each interval until smooth and just melted; do not overheat.
Stir the melted chocolate and 2 teaspoons vanilla into the warmed egg-and-sugar mixture until fully combined.
In a chilled bowl, beat 1 cup heavy cream with 2 tablespoons confectioners' sugar until soft peaks form; set aside.
In a separate large bowl, beat the unsalted butter until light and fluffy (about 2–3 minutes). Add the chocolate-egg mixture and beat on high for about 5 minutes until smooth and aerated.
Fold the whipped cream (from step 5) into the chocolate-butter mixture just until combined, taking care not to overmix.
Pour the filling into the cooled prepared pie crust, smooth the top, then refrigerate for at least 6 hours or overnight until fully set.
Before serving, make the topping by beating 1 cup heavy cream with 2 teaspoons instant vanilla pudding mix until soft peaks form; spread or scoop the whipped topping over the chilled pie and sprinkle with coarsely chopped milk chocolate.