In the bowl of a stand mixer, add the lukewarm milk and yeast. Let sit for 5 - 10 min until frothy. If you don't see any signs of yeast activity, your yeast may be dead. Throw out and try again with fresh yeast.
Add the flour, eggs, egg yolk, sugar, and salt. Fit the stand mixer with a paddle attachment and mix on low speed for a few minutes until the dough has come together. While the stand mixer is running, add the softened butter in pieces, waiting in between each piece. Continue mixing the dough on low speed for 15 - 20 minutes until smooth and passes the window pane test.
Place in a lightly greased bowl and let rise for about 2 hours, until just doubled in size. The dough will rise very slowly.
Once the dough has risen, gently punch it down to release the air. Shape into a ball and place back in the bowl. Cover and place in the fridge to rise for at least 8 hours and up to 48 hours.
When you are ready to shape the dough, prepare the cheesecake filling. In a medium bowl, use a hand mixer or stand mixer fitted with paddle attachment to beat the softened cream cheese on medium speed. Add the vanilla, lemon juice, lemon zest, chopped basil, and egg yolk. Beat until incorporated, scraping down the sides of the bowl as needed. Add the sugar and beat until incorporated. Set aside.
Grease a 12 cup muffin tin. Remove the dough from the fridge. It should have risen slightly in the fridge. Place on a floured surface and gently knead a few times to release the air. The dough should feel very firm now. Divide into 12 equal pieces (about 51g each, if you want to be very precise). Roll each piece into a ball and use your hands to gently stretch each piece into a circle about 4 inches in diameter. Place each piece of dough into a muffin tin so that it forms a cup.
Divide the cheesecake filling evenly among the 12 pastries. There should be 1 - 2 tbsp of filling for each.
Let rise for another 30 minutes until the dough is puffy. When you press a finger in, it should gently rise back about halfway. If it comes back all the way, it needs more time. Towards the end of the rise, preheat the oven to 400°F.
Whisk together the egg white and milk to create an egg wash. Brush the edges of the pastries with the egg wash. Top the middle with sliced strawberries, pressing each slice firmly into the center. As the pastries bake, they will expand, and the strawberries will take up less room then you think they will.
Place the buns in the middle rack and bake for 18 - 20 minutes until light golden brown and they make a hollow noise when tapped. Be careful not to overbake, as this will dry out the pastries.
Remove from the oven and let cool for 10 minutes. Then, gently release from the tin.
Store leftover pastries at room temperature in an airtight container. Over time, the basil and lemon flavor will become more pronounced.