In a medium sauce pan, heat milk and cream over medium heat until steam appears and milk is warm, about 5 minutes.
Combine the sugar, cocoa, instant espresso, eggs, and egg yolks in the bowl of a mixer and mix on medium speed until thickened, somewhat like cake batter.
Measure 1 cup of the hot milk/cream mixture and slowly add it to the eggs and sugar while the mixer is running on low speed. Mix until the 1 cup of milk/cream is completely incorporated.
Stir the chopped chocolate into the remaining hot milk/cream in the saucepan then add the egg mixture into the saucepan; stir to melt chocolate and combine ingredients. Cook over med-low heat, stirring gently and consistently, until the custard begins to thicken and coats the back of a spoon, 10-12 minutes. Be careful to not boil or the eggs could curdle.
When the custard is ready, remove from heat. Set a fine mesh strainer inside a medium bowl or large measuring cup, set in a bowl containing ice and water. (Using a bowl with a pour spout or a large measuring cup - 4 cups or larger - will make it easier to pour the custard once it is cooled). The custard will continue to thicken once cooled.
Strain the custard into the bowl set in the ice bath. (Be careful that the bowl is set firmly in the ice bath and not floating so that water cannot slosh into the custard.) Stir the vanilla into the custard then allow to cool to room temperature, stirring occasionally (about 30 minutes). (It is important to cool the custard to room temperature so it doesn't sweat once it goes into the refrigerator.)
Cover the custard with plastic wrap and refrigerate at least 3 hours or until completely cold (40 degrees or lower). At the same time, place the bowl or container you will use to mix and store the ice cream into the freezer to chill.
While the custard is cooling, mix the peanut butter, honey, and salt with a mixer on medium speed until smooth and somewhat fluffed. Refrigerate until ready to churn the ice cream.
Once cooled, pour the custard into an ice cream machine and churn, according to the manufacturer's instructions. While ice cream is churning, remove the peanut butter from the refrigerator and cut into chunks.
Transfer the ice cream to the chilled storage container, layering the chunks in the ice cream or gently fold them in -- work quickly to prevent the ice cream from melting. Cover and place in the freezer until firm, at least two hours.
Serve once the ice cream has hardened to the desired consistency.
Recipe yields 1.5 quarts (48 ounces) of ice cream.