Put all ingredients except the thyme and oil in a small food processor or mixing bowl. Whisk briskly or turn on the processor. Drizzle in the oil until a mayonnaise is formed. Then fold in the thyme. Alternatively, you could just use mayonnaise instead of the eggs and oil. Just fold in the other ingredients.
For the fritters: Heat olive oil in a pan and saute the onions and mushrooms until they are soft. Add the sherry and reduce until it has all been absorbed. Set aside to cool.
Make the fritter batter which is a Pate a choux: Heat milk and butter in a heavy gauge pot. Bring to a boil, then back off the heat just a bit. Use a wooden spoon to mix in the flour. Stir constantly to prevent burning. Cook until most of the moisture is out. The batter will look like a roux for sauces.
Transfer the "batter to a mixer and add in the eggs one at a time. Add in the herbs and seasoning at this time.
Fold the pate a choux into the mushrooms a little at a time. You want a good proportion of mushroom to batter; otherwise, it could become all puff and no mushroom. If you have leftover batter, you can bake it to make cream puffs, profiteroles, or eclairs. Also, fold in the thyme and goat cheese at this time so they will stand out. And, adjust seasoning as needed.
Heat the frying oil either in a heavy duty pot or a fryer to 350 degrees. Scoop fritter batter into the oil and allow to rise to the top. If it gets stuck on the bottom, knock it loose with a spider or spoon. It will brown on one side. To create an even cook, either place a fryer basket over the fritters to keep them submerged or flip them occasionally with a "spider" or slotted spoon.
When they are golden brown and cooked through, remove from the oil onto paper towels to absorb excess oil. Serve with aioli for dipping.