Cook sugar and water in a small saucepan over medium-high heat, gently swirling the pan (do not stir), until the mixture turns golden brown, 5 to 7 minutes. Immediately, pour the caramel carefully into a 9-inch round metal cake pan. If it stiffens before it completely covers the bottom of the pan, warm the pan in the oven, then swirl to cover the bottom.
Beat cream cheese in a large bowl with a mixer at medium-high speed until soft. Beat in eggs, one at a time, until thoroughly combined, scraping down the sides. Add condensed milk, evaporated milk and vanilla; mix until combined. Carefully pour into the prepared pan. (You may hear some cracking; the pan will be very full).
Place the cake pan in a larger pan and pour enough hot water around it to come 1 inch up the sides. Bake until golden and set at the edges but still wobbly at the center, about 1 hour. A knife inserted in the center should come out clean.
Transfer the cake pan to a wire rack and let cool to room temperature, about 1 hour. Then cover and refrigerate for at least 4 hours or overnight. To serve, run a knife around the edge of the pan and invert the flan onto a plate.