Generously butter soufflé dish, then sprinkle with Parmesan cheese, tapping the dish lightly on the counter and knocking out excess.
Stir together cauliflower, parsley, 1/4 teaspoon salt and white pepper to taste in a large bowl.
Melt butter in a heavy saucepan over medium heat.
Whisk in flour and continue whisking, until pale golden.
Mix the milk and water together and add a little at a time, whisking constantly until very smooth. Bring sauce to a boil, whisking, then simmer until quite thick, about 1 minute- make sure you never let it rest- keep stirring.
Remove from heat and hand whisk in yolks, 1/4 teaspoon white pepper, and 1/2 teaspoon salt.
Stir into cauliflower mixture.
Beat whites in a bowl with an electric mixer at high speed until they just hold stiff peaks (they should not look dry, just stiff). Stir a heaping spoonful of whites into yolk mixture to lighten, then gently fold in remaining whites until just combined.
Gently pour into soufflé dish (leave at least 1 inches of space at top) and bake until golden brown and top appears set, 35 to 40 minutes.
To Make Brown Butter
Cook butter in a heavy saucepan over medium heat, stirring occasionally, until it turns golden with a nut-like fragrance and flecks on bottom of pan and turn a rich caramel brown, about 6 minutes. Remove from heat.
Serve soufflé immediately, drizzling with warm brown butter just before you serve.