Adjust oven rack to lower-middle position and preheat oven to 400°F (205°C). Line a rimmed baking sheet with parchment paper. Set aside.

In a large skillet, heat 2 tablespoons sun-dried tomato oil over medium until shimmering. Add sun-dried tomatoes, carrots, mushrooms, and salt, and cook, stirring often, until carrots and mushrooms are softened, liquid evaporates, and mixture appears dry, 4 to 5 minutes. Remove from heat and let cool for 10 minutes.

Stir feta cheese, mixed herbs, olives, walnuts, scallions, crushed red pepper, black pepper, and 1 egg into the cooled carrot mixture in skillet.

In a small bowl, whisk together milk, remaining 1 egg, and remaining 2 tablespoons sun-dried tomato oil.

Working in batches of 6 phyllo sheets for 1 spiral (keep remaining sheets covered), place 1 phyllo sheet on a work surface with long end of sheet running parallel to edge of counter. Using a pastry brush, lightly brush egg mixture along 1 short edge of sheet. Put a second phyllo sheet overlapping the short edge by 1 inch with long end of sheet running parallel to edge of counter. Press to seal both sheets together to form a sheet about 26 inches long. Lightly brush top with egg mixture (some dry spots are OK). Place 2 more phyllo sheets on top, overlapping the short edge by 1 inch. (Don’t brush the overlapping center or it will get too soggy.) Repeat brushing the top lightly with egg mixture. Place 2 more phyllo sheets and repeat overlapping in center and lightly brushing the top.

Spread about 1 cup cooked carrot mixture along bottom (long edge) of phyllo, leaving a 1-inch border from bottom edge. Fold bottom edge of phyllo over filling and continue rolling phyllo away from you into a log (do not roll tightly).

Starting at 1 end of the log, coil the end around itself, creating a spiral shape. Continue to wrap the log around, coiling tightly. Transfer to center of prepared baking sheet. Cover with a clean kitchen towel.

Repeat steps 5 and 6 with remaining 24 phyllo sheets and remaining 4 cups filling to create a total of 4 more logs. Attach 1 log to the end of the first spiral on baking sheet, and wrap the log around to continue spiral. Repeat three more times to create 1 large spiral on baking sheet.

Lightly brush pastry with egg mixture (do not let egg mixture pool), taking care to coat seams where pastry logs connect to make spiral. Sprinkle with everything bagel seasoning or sesame seeds.

Bake until well browned, 25 to 30 minutes. Let cool 10 minutes before transferring to a cutting board. Cut into wedges and serve hot.