In a small bowl, whisk the mayonnaise, chives, lemon juice, mustard, thyme, cayenne, salt, pepper, garlic and lemon zest together to blend. (Sauce can be prepared 1 day ahead. Cover and refrigerate.)
In a medium bowl, whisk together the flour, chives, thyme, salt and pepper. Pour the chilled beer into the flour mixture and whisk until the batter is smooth.
Pour the oil into a heavy large wide saucepan. Attach a deep-fry thermometer to the side of the saucepan. Heat the oil over medium-high heat until the temperature registers between 330 and 350 degrees F. Working in batches, dip a few mushrooms at a time into the beer batter to coat generously, allowing any excess batter to drip back into the bowl. Drop the mushrooms, a few at a time, into the hot oil and fry until the batter is deep golden brown and the mushrooms are tender, adjusting the heat as needed to maintain the oil temperature between 330 and 360 degrees F, 3 1/2 to 4 1/2 minutes per batch. Using a slotted spoon, remove the mushrooms and drain on paper towels.
Mound the mushrooms on a serving platter and place the dipping sauce alongside. Serve while the mushrooms are still hot.