Preheat the oven to 400°F
Place the strawberries in a large bowl. Add the olive oil and a 1/4 cup of the sugar. Toss to coat. Transfer the berries to a baking sheet lined with parchment paper. Roast the berries for 15-20 minutes or until lightly browned.
Transfer the berries, juice and all, to a fine mesh strainer. Force any remaining juice out by pressing the berries with a muddler or the back of a spoon. Set the juice and the mashed berries aside to cool completely.
Pour the milk into a small saucepan and place over low heat. Heat the milk until steam settles on the surface. Remove from the heat.
In a medium-sized bowl, whisk to combine the egg yolks, sugar and salt. Whisk until the sugar is mostly dissolved and the egg yolks turn a paler shade of yellow.
Pour half of the warmed milk into the yolk mixture, whisking constantly. Pour the yolk mixture into the saucepan with the rest of the milk and return the pan to the heat. Cook the custard until it thickens slightly, about 8 minutes.
Remove the mixture from the heat and pour into a large bowl. Whisk in the whipping cream and the strawberry juice. Transfer the mixture to the fridge and chill for 1 hour.
Pour the chilled mixture into the bowl of an ice maker and churn for 30 minutes. In the last five minutes of churning, spoon in the berry mush until it is swirled throughout. Spoon the finished ice cream into a 8x8 pan or loaf pan and cover. Freeze overnight.
The next morning, place the ice cream on the counter for 15 minutes to soften. Spray a 6" bowl with non-stick spray and line with plastic wrap. Spoon the softened ice cream into the bowl and smooth the surface with the back of a spoon. Cover the surface and return the ice cream to the fridge. Let set overnight.