Separate the eggs, putting the whites in one clean mixing bowl and the yolks in another. Add ½ cup of the sugar to the yolks and beat with an electric mixer on medium-high speed until pale yellow, 2 to 3 minutes. On medium speed, beat in the mascarpone and cream cheese until smooth, 1 minute.
Whip the egg whites on medium-high speed with clean beaters until foamy, about 2 minutes. Add ¼ cup of the remaining sugar and beat until the whites form stiff peaks when the beaters are lifted, about 2 minutes more. Gradually fold the whites into the mascarpone mixture to create a light-textured mousse. Cover and refrigerate it.
In a medium saucepan, stir together the water and remaining ¾ cup of sugar. Cook over medium-high heat until the mixture reaches 218ºF on a candy thermometer, 4 to 6 minutes. Stir in the hulled strawberries and whole sprigs of basil and cook until the berries soften and the syrup is pink and lightly thickened, 3 to 4 minutes. Remove the pan from the heat and use a slotted spoon to place the strawberries onto a sheet of foil, letting the excess syrup drip back into the pan. Let the syrup cool slightly. Discard the basil.
To assemble the tiramisu, spread a thin layer of mousse in a 2-quart serving dish. Soak the ladyfingers in the warm strawberry-basil syrup, turning once or twice with tongs, until softened and saturated, 4 to 5 minutes. Over the mousse, layer half of the ladyfingers followed by another layer of mousse. Then layer on the strawberries, remaining ladyfingers, and remaining mousse. Cover and refrigerate the tiramisu for at least 30 minutes or up to 3 days.
Before serving, sprinkle it with cocoa powder, and garnish each portion with some sliced fresh strawberries and chopped fresh basil.
Tip: If you’re wary of using uncooked eggs, you can pasteurize them yourself (most eggs aren’t pasteurized). To do it: Place the eggs in a saucepan and cover them with water. Clip a thermometer to the pan, place the pan over low heat, and bring the water to 140ºF (but not over 142ºF). Keep the water at that temperature for 3 minutes, adjusting the heat or moving the pan off the heat as necessary. Then place the eggs under cold running water to stop the heating. Use them immediately or refrigerate.