Line a baking sheet with parchment paper. In a heavy saucepan over medium heat, melt the butter with the sugar, corn syrup, water, and salt, stirring until the sugar dissolves.Keep cooking, stirring often, for 8 to 12 minutes, until the toffee turns deep amber and reaches 300°F on a candy thermometer. If the butter starts to separate, whisk vigorously until it comes back together, then pour the toffee onto the parchment and spread it thin.
Scatter the chocolate chips over the hot toffee, wait 2 minutes for them to soften, then spread into an even layer. Let it cool until hard, refrigerate for 20 minutes to set the chocolate, then place it in a zip-top bag and tap with a rolling pin to break it into small bits.
In a medium saucepan, whisk together the granulated sugar, brown sugar, espresso powder, cornstarch, and salt. Whisk in the coconut milk and corn syrup, then cook over medium heat, whisking constantly, for 5 to 7 minutes, until the base simmers and thickens slightly.
Remove from the heat and stir in the vanilla. Pour into a bowl, press plastic wrap onto the surface, and refrigerate for at least 6 hours or overnight.
Pour the chilled coffee base into an ice cream maker bowl that has been frozen for at least 24 hours. Churn for 20 to 25 minutes, until thick and creamy.
Quickly fold about three quarters of the toffee bits into the churned ice cream, saving the rest for topping.
Spread into a 9x5-inch loaf pan, sprinkle with the remaining toffee, cover tightly, and freeze for 4 to 6 hours, until firm. Let sit at room temperature for 10 minutes before scooping.