Make Soufflé: Preheat oven to 400°F with rack in middle.
Generously butter soufflé dish, then sprinkle with cheese, knocking out excess.
Stir together cauliflower, parsley, 1/4 teaspoon salt, and white pepper to taste in a large bowl.
Melt butter in a 2- to 3-quart heavy saucepan over medium heat. Whisk in flour, then cook, whisking, until pale golden, about 2 minutes. Add milk a little at a time, whisking constantly until very smooth. Bring sauce to a boil, whisking, then simmer, whisking, until quite thick, about 1 minute. Remove from heat and 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). Stir a heaping spoonful of whites into yolk mixture to lighten, then gently fold in remaining whites until just combined.
Gently spoon into soufflédish (leave at least 1 inches of space at top) and bake until golden brown and top appears set, 35 to 40 minutes.
Make Brown Butter: Cook butter in a small heavy saucepan over medium heat, stirring occasionally, until it turns golden with a nutlike fragrance and flecks on bottom of pan turn a rich caramel brown, about 6 minutes. Remove from heat.
Serve soufflé immediately, drizzling with warm brown butter.