Add the milk, cream, ⅓ cup of the sugar, espresso, kahlúa and vanilla to a medium-large saucepan. Stir to blend and place it over medium heat. Cook just until the sugar has completely dissolved, and it's beginning to simmer, a few minutes. Turn off the heat and leave this on the stove.
In a medium-sized mixing bowl, combine the egg yolks with the remaining 1/4 cup of sugar. Use an electric mixer, with the whisk attachment, on a high speed to blend until the eggs become thick and pale yellow, about 3 minutes. (You can of course also use a hand whisk -- it will just take longer and use a lot of muscle.)
Pour about 1 cup of the warm milk mixture into the egg mixture and immediately stir to blend. Then add this mixture back into the saucepan.
Cook over very low heat, stirring constantly (a wooden spoon is best), until the mixture becomes thick enough to coat the back of a wooden spoon, about 7 minutes. (Timing is important here -- you want to remove it as soon as it's at the point -- usually it's when I see the first couple of bubbles from the heat. If it goes too long it might curdle.)
Place a strainer over a medium-sized bowl and pour the warm custard mixture through the strainer and into the bowl. Once it's at room temperature, cover tightly with plastic wrap and refrigerate for at least 4 hours and ideally overnight.
Transfer to an ice-cream/gelato maker, and freeze according to the machine's instructions.
While the it's churning, crush the chocolate covered espresso beans. Here's how: add them to a heavy-duty, zip-lock bag and seal it, removing the air. Then use a meat mallet or the bottom of a small, heavy-bottomed sauté pan to crush the beans.
Once the gelato is ready, gently mix in the crushed chocolate-covered espresso beans. Serve immediately if you want it very soft, or place it in the freezer until you're ready to serve.
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