Add 350g cream cheese to a large mixing bowl, use a spatula to cut the cream cheese into smaller cubes to speed up the softening time. Leave the cream cheese and 4 eggs on the countertop for about 30 minutes until they reach room temperature. Poke the cream cheese to check the softness, it should be soft like toothpaste. Preheat the oven to 420 °F.
Use the lowest speed (speed 1) of a hand mixer to whip the softened cream cheese for 1 minute, add in 70g sugar, mix with the hand mixer for about 30 seconds until the cream cheese looks smooth. Crack 3 room-temperature eggs into the whipped cream cheese one by one. Every time you add in an egg, mix with the lowest speed of the hand mixer for about 30 seconds until all the egg liquid has been absorbed into the cream cheese before adding the next egg.
Add 1 egg yolk and 1 tsp of vanilla extract (optional) to the mixture, mix with the hand mixer for 10 seconds until there's no yellow egg yolk mark in the mixture.
Add 175g heavy cream to the mixture, mix with the lowest speed of your hand mixer for 30 seconds.
Add 10g sifted cornstarch (or cake flour) to the mixture, mix with the lowest speed of your hand mixer for 15 seconds.
Place a strainer over a large bowl, pour the mixture through the strainer to filter out any lumps. Grease a 7 Inch non-stick springform pan with a small piece of butter (or brush with neutral cooking oil). Rinse a large piece of parchment paper with cold water, squeeze and shake off the excess water. Line the buttered cake pan with parchment paper. Cut off the extra parchment paper so that the edge of the parchment paper comes about 2" above top of pan on all sides.
Pour the Boba Basque Cake mixture to the cake pan. Place the cake pan on the upper middle oven rack to bake at 420 °F for 25 to 28 minutes.
Once the top of the cheesecake looks caramelized brown and the edges are set, take out the Basque cheesecake from the oven to let cool completely at room temperature.
Cover the cooled Basque cheesecake with food wrap, chill it in the fridge for at least 4 hours, ideally over night.
Unmold the Basque cheesecake from the cake pan, peel off the parchment paper, place the cheesecake on a serving plate. Optionally serve with brown sugar boba cream sauce on the top, see content below.