Position a rack in the center of the oven and preheat oven to 375 degrees F.

Heat 1 tablespoon olive oil in a large pan over high heat. Add the mushrooms and saute until tender and they have released their liquid. Remove from the pan, draining all the liquid. To the same dry pan, add the spinach and cook until just wilted, about 3 minutes. Drain, squeeze out the excess water, and coarsely chop. Combine the sauteed spinach and mushrooms with the roasted red peppers, roasted garlic pulp, oregano, red pepper flakes, and salt and pepper, to taste. Gently fold in the goat cheese.

Stack the sheets of phyllo dough on top of each other and cut them into thirds. You will have a total of 12 sheets. Divide the sheets into 4 (3-sheet) stacks. Lightly brush the top sheet of each stack with a teaspoon of olive oil. Put 1/4 cup of the spinach mixture on the bottom edge of each oiled sheet. Then, beginning at the bottom, roll up the phyllo securely; when you have rolled the filling into the center of the phyllo, at that point, fold in the ends and continue to roll tightly, pressing to close. Lightly brush each strudel with the remaining oil. Line a baking sheet with parchment or waxed paper. Place the strudels on the paper about 2-inches apart and bake until golden brown, about 15 minutes. Remove from the oven. Cut each strudel in half diagonally and serve warm.