Source: HowToCook (a programmer's guide)
Heat your oven to 220°C. The aggressive temperature is the point — it creates the characteristic burnt exterior and barely-set, almost liquid centre that defines a Basque cheesecake. This isn't a gentle bake.
Warm the cream cheese in short 10-second blasts in the microwave until it yields to a spatula without lumps. Don't overshoot; you need it softened, not hot. Add the caster sugar and whisk with an electric mixer until the mixture is pale and aerated — roughly two minutes. This aerating step matters: it traps air bubbles that will expand under heat-control|high heat and give the cake its characteristic wobble and slight collapse on cooling.
Add the eggs and yolk, whisking until fully incorporated, then pour in the double cream. The cream prevents the filling from curdling under the intense oven temperature. Sift the flour over the batter and fold it through gently until no streaks remain. If using chocolate, grate it finely and fold it in last. The flour keeps the filling from splitting, though a Basque cheesecake is meant to look alarmingly underbaked — that's not a mistake.
Pour into a loose-bottomed tin lined with baking parchment (the paper prevents the base from sticking whilst allowing air to circulate underneath). Bake for 20–22 minutes. The cake will puff dramatically and the edges will blacken. The centre should wobble conspicuously when you shake the tin — this is correct. The temperature gradient from the scorched rim to the molten middle is what you're engineering.
Cool in the tin to room temperature, then refrigerate overnight. The overnight rest is non-negotiable: dairy|the proteins in the cream cheese continue to set gently and the cake firms to a texture that's fudgy, not liquid. Serve straight from the fridge. The contrast between the bitter caramelised crust and the cool, yielding centre is the entire dish.
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