Allow the grated white cheddar cheese to reach room temperature and toss it with flour to promote smooth melting. In a bowl, mix half and half, white wine, Worcestershire sauce, dry mustard, pepper, and mayonnaise or sour cream. Set this mixture aside while you prepare the mushrooms. Warm it gently using a double boiler over simmering water for about 15 minutes, stirring periodically. Gradually incorporate the cheddar cheese and stir until fully melted and integrated. Remove from heat once the sauce is creamy.

For presentation, the sauce can be spread on a serving platter with the mushrooms arranged on top, lightly drizzled over the mushrooms, or served separately in ramekins for dipping.

Begin by preheating the oven to 375°F. Clean the mushrooms using a damp paper towel. Carefully remove the stems with a paring knife; reserve the stems for future use if desired. Fill each mushroom cap nearly to the brim with the herb pub cheese.

In a separate bowl, combine Parmesan cheese, chopped provolone, and half and half. Microwave this mixture for 25 seconds, then incorporate the Panko breadcrumbs. Let it sit briefly to cool and solidify slightly, which aids in application. Top each filled mushroom cap with a portion of this mixture using your hands for even distribution.

Position the mushrooms on a baking sheet lined with foil or parchment paper. Bake for approximately 15 minutes, or until the topping is golden and the mushroom bases are firm. Rotate the pan midway through for uniform browning if necessary.

Allow the mushrooms to rest for 5 minutes before serving to settle their structure and temperature.