In a medium saucepan, combine the whole milk or half-and-half, heavy cream, half of the sugar, medium-ground coffee, and kosher salt. Heat over medium heat until hot but not boiling, about 5–7 minutes.

Remove from heat and let steep for 15 minutes.

In a medium bowl, whisk egg yolks and remaining sugar until well combined and slightly thickened, about 2 minutes.

Fill a large bowl with ice and cold water to create an ice bath. Set a fine mesh strainer over a medium-sized bowl.

Gradually pour the warm coffee mixture into the bowl of egg yolks while whisking continuously. Return to saucepan over medium heat, stirring constantly until thickened, about 5–7 minutes.

Strain the custard through the fine mesh strainer into the bowl set in the ice bath. Stir occasionally until cool, then refrigerate for at least 4 hours or overnight.

Churn the cooled custard in an ice cream maker according to the manufacturer's instructions, layering in cajeta and chopped chocolate.

Transfer churned ice cream into an airtight container, smoothing the top, covering with parchment paper, and sealing. Freeze for at least 4 hours.