Preheat the oven to 475 degrees F. Heat a large sauté pan over medium heat. Add the olive oil and heat. Add the garlic, shallot and mushrooms and cook, stirring often, until almost tender, about 4 minutes. Add the spinach and cook until very tender, about 6 minutes. Transfer the mixture to a bowl and fold in the goat cheese; season with salt and let cool.
Lay out 1 phyllo sheet, butter liberally and fold in half to form a rectangle. Butter again and fold in half again to form a skinnier rectangle. Repeat with the remaining phyllo, placing a towel over the prepared sheets to prevent them from drying out.
Pile the mushroom and spinach mixture on one end of each phyllo rectangle. Fold up one corner of the dough over the filling to form a triangle and continue to fold (like a flag), buttering between each fold. Transfer to a parchment-lined baking sheet and bake until golden brown, 10 to 15 minutes.