1. Cook the strawberries over a medium-high flame, first mashing with a spoon or potato masher to help squeeze the liquid out faster (how well they are mashed doesn’t matter, as you will purée them later). Continue to cook while stirring occasionally until the liquid is cooked out and a wooden spoon leaves a trail, about 15 minutes. As the liquid is reduced you will need to stir more frequently to prevent scorching. Set aside to cool.
2. Beat the egg whites, salt, and lemon juice until soft mounds appear. Gradually add one cup of the sugar and continue beating until the egg whites are glossy and hold stiff peaks. Transfer to an extra-large mixing bowl.
3. In a food processor, process the egg yolks, remaining sugar, cooked strawberries, and bananas until light and fluffy. With the food processor still running, drizzle the oil in slowly until it is fully incorporated. Transfer to a mixing bowl and whisk in one fourth of the beaten egg whites to lighten the mixture.
4. Add the strawberry mixture to the rest of the beaten egg whites and fold the two mixtures carefully together. Freeze immediately and serve frozen.