Place the mushrooms in the bowl of a food processor fitted with a steel blade and pulse until finely chopped. Remove, set aside, and lightly wipe the food processor clean. Fit food processor with a grater blade and grate potatoes into long, thin strips. (Alternatively, you can finely chop the mushrooms and coarsely grate the potatoes by hand.)

In a large bowl, lightly beat the eggs with 1/2 teaspoon kosher salt and 1/4 teaspoon pepper. Fold in the grated potatoes until evenly combined. Set aside.

Heat 2 tablespoons olive oil in an 11 or 12-inch skillet over medium high heat. Add mushrooms and cook for 6 minutes, stirring often, until the mushrooms are soft and the mushroom liquid has evaporated. Add the lemon juice, tarragon, garlic, remaining 1/4 teaspoon salt and 1/4 teaspoon pepper, and cook 30 seconds. Add spinach and cook just until the spinach begins to wilt, about 1 minute more. Stir in half of the feta, then transfer from skillet to a plate and set aside.

In the same skillet you sautéed the mushrooms, heat 3 tablespoons olive oil over medium. Spread half of the potatoes over the bottom of the skillet in an even layer, pressing them gently, then top with the mushroom filling and remaining feta cheese. Gently press remaining half of the potatoes over the top of the filling to create a “lid.” Let cook on medium until the bottom potato layer is crispy and golden, 5 to 6 minutes.

Run a heat-proof spatula between the potato dish bottom and skillet to loosen it (a long fish spatula works well). Quickly flip the potato skillet out and over onto a large, clean plate so that the cooked side is “up”. Return skillet to heat, add remaining 3 tablespoons olive oil, then slide the potato dish back into the skillet, uncooked side down. Let cook until the bottom of the dish is golden and the potatoes are tender, 6 to 10 additional minutes. Slide onto a plate, cut into wedges, and serve.