Pulse the mushrooms in a food processor until very finely chopped (or finely chop with a knife). Heat 1 tablespoon olive oil in a large nonstick skillet over medium-high heat. Add the mushrooms and season with salt and pepper. Cook, stirring occasionally, until the liquid is released and the mushrooms are lightly browned, about 5 minutes. Add 2 tablespoons dill and half the garlic; stir to combine and remove from the heat. Scrape into a large bowl and let cool, stirring occasionally, about 10 minutes. Wipe out the skillet.
While the mushrooms cool, grate the beet on the large holes of a box grater. Toss with the remaining 1/4 cup dill, the horseradish, pickle brine and a pinch each of salt and pepper. Mix the sour cream with the remaining garlic; season with salt and pepper.
Add the turkey and a big pinch each of salt and pepper to the cooled mushrooms; mix well. Form into four 4-inch patties. Heat the remaining 1 tablespoon olive oil in the reserved skillet over medium-high heat. Add the patties and cook until well browned, about 4 minutes per side.
Spread the sour cream sauce on the buns. Serve the burgers on the buns with lettuce, tomato, onion and the beet slaw. Serve with the pickle spears.
Make a quick slaw with raw beets -- just grate them on the large holes of a box grater. Bonus: Beets are a good source of fiber and vitamin C.