Toss the squash cubes in the olive oil and spread them out on a baking sheet. Roast until the squash is tender and starting to caramelize, 45 to 50 minutes. Set aside to cool.
Once the squash has cooled slightly, transfer it to a food processor and puree until smooth.
In a large saucepan, combine the cream, milk, honey and 2 tablespoons of the brown sugar, and bring to a mild simmer over medium heat.
In a bowl, whisk the egg yolks vigorously until they turn pale yellow. Add the remaining 6 tablespoons brown sugar and stir to fully incorporate. Temper the eggs by slowly adding one third of the heated cream while whisking continuously.
Pour the tempered egg mixture back into the saucepan with the rest of the cream and cook over low heat (do not boil!), stirring constantly with a wooden spoon, using figure-eight motions. Cook until the custard begins to thicken and coat the back of the spoon, anywhere from 15 seconds to 1 minute.
Pour the custard through a fine mesh sieve into a clean bowl set over an ice bath. Stir until the custard is completely cooled.
Combine a third of the cooled custard with 1/2 cup of the squash puree (doing this in a small batch will ensure a smoother texture), and use an immersion blender to blend until smooth. Add the custard-squash mixture back into the remaining custard. Stir in the curry powder, salt, cumin, cloves and black pepper. Blend and emulsify with the immersion blender until smooth. Strain, then cover and refrigerate for 4 hours to overnight.
Pour the custard into the bowl of an ice cream maker and churn according to the manufacturer's directions. Transfer to a container suitable for the freezer and freeze for at least 2 hours before serving.