Mix the yogurt, goat cheese, sundried tomatoes, finely chopped basil, and minced garlic in a bowl until combined. Set it aside.
Set up the mushroom breading station: flour, beaten egg, breadcrumbs, and cleaned Portobello mushrooms. Make sure to season the flour, eggs, and breacrumbs with a pinch of salt and pepper.
Roll the mushrooms first through the flour, then the egg wash, and finally, the Panko breadcrumbs. Feel free to use your hands to make sure the mushrooms are well-coated in each layer.
To a pan over medium-high heat, add the oil. When oil is hot, add the breaded mushrooms and cook until golden on one side (about 4-5 minutes). Then, flip and cook them on the other side until golden and cooked-through (another 4-5 minutes). If you’re deep-frying the mushrooms in a deep fryer where the mushrooms are completely submerged in oil, you’ll only need about 5 minutes total.
Once the mushrooms are fried, remove them to a plate with a paper towel to absorb the extra oil. Season them immediately with a sprinkle of salt (ideally with larger flakes like Maldon sea salt or kosher salt).
Optionally, toast the brioche buns. If using brioche buns, you can just toast them for a couple minutes in a dry pan over medium heat. You don’t need butter as there’s plenty of butter in brioche buns.
Finally, assemble the burgers. Add about a tablespoon of the goat cheese mixture to the underside of the top and bottom buns each. Then, add 2 fried mushrooms with a bit of goat cheese mixture in between to hold them together. Top the second mushroom with the remaining goat cheese mixture (if you have any left). And finally add some arugula or greens on the top of the goat cheese on the top bun.