Preheat oven to 400° F. Butter the inside of a 2-quart soufflé dish and sprinkle with the Parmesan cheese, knocking out the excess. Stir together cauliflower, parsley, salt and pepper in a large bowl.

In a heavy-bottomed saucepan, melt 1/4 cup butter and whisk in ¼ cup flour to make a roux. Cook the roux for about 3 minutes until it no longer smells like raw flour. Add milk slowly, whisking until smooth. Thicken the sauce for about 1 minute. Remove from the heat and whisk in the egg yolks, one at a time, until all the yolks have been added and the mixture is smooth and creamy. Stir the thickened egg and milk mixture into the cauliflower mixture.

Beat whites with an electric mixer on medium-high speed until stiff peaks form. Gently fold whites into the cauliflower mixture until just combined. Gently spoon into prepared soufflé dish, leaving 1 inch of space at top. Turn down oven to 375° F and bake until golden brown and top appears set, 35 to 40 minutes. Cook the salted butter in a heavy-bottomed saucepan over medium heat. Stir occasionally until it turns golden-brown and has a nutty fragrance and flecks on the bottom of the pan turn caramel-brown, about 6 minutes. Serve soufflé drizzled with brown butter.