Hull and slice the strawberries. Toss them with a drizzle of olive oil and a drizzle of honey. Line a baking sheet with parchment paper.
Roast the strawberries on the baking sheet for 20 minutes.
Transfer the strawberries to a bowl and add the lemon juice and one tablespoon of maple syrup. Gently smash the strawberries, but allow some chunks to remain.
Heat a pot of water over medium-high heat. Place a ceramic bowl over the hot water to create a double boiler, and heat the coconut milk in the bowl until hot (not boiling). Whisk in the remaining maple syrup (2 tablespoons).
In another bowl, whisk 3 egg yolks. Add 1 ladleful of the hot coconut milk to the egg yolks while whisking vigorously.
While whisking constantly, repeat Step 6 twice more to make a total of three ladlefuls of hot coconut milk.
Pour the egg-coconut milk mixture back into the double boiler and whisk for 2-3 minutes nonstop, until the custard has thickened slightly.
Cover the custard with plastic wrap and cool it completely. Stir in the smashed strawberries.
Pour the custard into an ice cream maker and churn according to the manufacturer’s directions.*
Alternatively, you may pour the custard into a shallow dish and freeze. Stir it every 5-10 minutes, until the mixture is soft and scoop-able.