Combine yeast, 2 tsp caster sugar, milk and 1/3 cup (80ml) lukewarm water in a jug. Stand in a warm place for 10 minutes or until frothy.
Meanwhile, place flour and remaining 1/3 cup (75g) caster sugar in a stand mixer fitted with the dough hook. Add yeast mixture and egg yolks, and knead until combined. With the motor running, add butter, 1 piece at a time, kneading well after each addition. Knead for 10 minutes or until butter is completely incorporated and dough is smooth and elastic (or knead by hand). Transfer to a greased bowl. Cover and set aside in a warm place for 1 hour or until doubled in size. Divide dough into 14 even pieces, roll into tight balls and place on a greased tray. Cover and set aside for 20 minutes or until risen slightly.
Meanwhile, to make the cheesecake filling, in the cleaned stand mixer fitted with the whisk attachment, whisk cream cheese, sour cream, 1 tbs icing sugar and vanilla until smooth. Transfer to a piping bag fitted with a 1.5cm plain nozzle.
To make the strawberry filling, combine strawberries and remaining 2 tsp icing sugar in a bowl and transfer to another piping bag fitted with a 5mm plain nozzle. Half-fill a large saucepan or wok with oil and heat to 160°C (a cube of bread will turn golden in 3 minutes when oil is hot enough). Place extra caster sugar on a plate. In batches, fry doughnuts, turning halfway, for 8 minutes to cook through. Toss hot doughnuts in extra caster sugar, then drain on paper towel. Insert a knife into each doughnut to create a hole.
Pipe in cheesecake filling, then strawberry filling. Scatter with ground dried strawberries, if using, to serve.