Line a loaf pan with parchment paper.
Add the quartered strawberries, sugar and lemon juice to a small saucepan and bring to a boil over medium-high heat. Once boiling, turn to medium-low and keep at a simmer for 5-8 minutes until cooked down into a jammy texture with just a few tablespoons of liquid left. Use a fork to crush the strawberries into a chunky texture. Remove from the heat and pour the strawberries into a bowl and place in the fridge to cool.
Add the finely chopped basil to a bowl with 1 tablespoon of sugar and 1 1/2 tsp of extra-virgin olive oil. Use your fingers to mix everything together to combine and then set aside.
In a medium mixing bowl, stir together the vanilla extract and sweetened condensed milk until combined.
In a large mixing bowl, use a hand mixer with whisk attachment to whip the heavy cream to soft-medium stiff peaks. Alternatively, whip the cream with a whisk attachment in a stand mixer to soft-medium stiff peaks.
Pour the sweetened condensed milk mixture into the whipped cream mixture and gently fold everything together to combine (it should be fairly smooth). Be careful not to overmix.
Next, add the cooled strawberry mixture to the whipped cream mixture and gently fold in to combine.
Pour half of the strawberry-whipped cream mixture into the prepared loaf pan and sprinkle over half of the fresh basil mixture. Use a knife to swirl it into the ice cream mixture. Pour in the remaining strawberry-whipped cream mixture and sprinkle over the rest of the fresh basil mixture. Use a knife to swirl it into the mixture again.
Cover tightly with plastic wrap and freezer for at least 8 hours or overnight before serving. If the ice cream is too difficult to scoop, let it warm up for about 5-10 minutes on the counter before scooping.
Store in the freezer, covered well with plastic wrap, for up to one month.