Preheat oven to 300°F. Line the base of an 8-inch round springform pan with parchment paper.
For the crust, combine flour, brown sugar and salt in a medium bowl so there are no lumps of brown sugar. Stir in melted butter and use your fingertips to combine all ingredients so it forms damp crumbs. Turn the crumbly mixture out into prepared pan and press it down in an even layer, pushing it about 1-inch up the sides to form an edge. Run the back of a spoon across the surface to smooth it out. Bake for 15-17 minutes until slightly puffed and lightly golden. Transfer to a wire rack.
For the filling, first infuse the cream or milk. Pour cream or milk into a small saucepan. Split the vanilla bean in half lengthwise then run a paring knife along the inside of it to scrape out all of the little black seeds. Add the seeds to the cream or milk and then place the scraped vanilla pod in there too (it also has lots of flavour!). Cover the saucepan and place it over medium-low heat until it just comes to a simmer. Turn off the heat and let it stand, covered, for 10 minutes to infuse.
Add cream cheese to a large bowl and beat using a hand mixer on medium speed until smooth. You can also use a stand mixer with the paddle attachment, or you can use a food processor. Add sugar and beat until silky and glossy. Beat in eggs one at a time, and mix well between each addition so each egg is evenly incorporated before adding the next one. Mix in sour cream until evenly blended. Gradually pour in the vanilla-infused cream or milk while mixing until evenly combined. If you're not using the vanilla bean, then this is where you can mix in the vanilla extract.
Pour the filling slowly over the pre-baked crust and smooth out the surface. Gently tap the pan on the counter to release any air bubbles, and then pop the bubbles that come to the surface. Bake for 45 minutes until set around the edges and slightly wobbly in the center. Turn off the oven and leave it in (with the door still closed) for another 15 minutes. This gentle cooking produces the silkiest texture. Open the oven door and let it cool in the oven for 10 minutes before transferring to a wire rack to cool completely. The gentle cooling will help to prevent cracking.
Once completely cool, make the topping. Combine sour cream with sugar until smooth then spread it over the cheesecake in a thin layer. Place the cheesecake back in a 350 degree oven for 8-10 minutes just until the topping looks set - it will lose its sheen and appear matte.
Refrigerate for at least 2 hours or overnight. When ready to eat, slice with a hot dry knife and wipe the knife clean between each slice.