Preheat the oven to 475°F. Using a small spoon, scrape the gills from the underside of the mushroom caps. Brush the mushroom caps lightly with olive oil (if you have olive oil spray, use it!) and season them generously with salt and pepper. Arrange them gill side down on a baking sheet and roast until tender, 10 to 12 minutes. Transfer the mushroom caps to a paper towel-lined plate to drain. Wipe any excess water from the baking sheet. (Leave the oven on.)

Heat 2 tablespoons of olive oil in a large skillet over medium heat. Add the panko, garlic, 1/4 teaspoon salt and 1/4 teaspoon pepper. Cook, stirring, until panko is toasted and garlic is fragrant, about 3 minutes. Add the spinach and cook until wilted, about 3 minutes. Remove from the heat and stir in the feta, lemon zest, oregano, and dill.

Arrange the mushroom caps gill side up on the baking sheet. Divide the spinach mixture evenly between them. Return the mushrooms to the oven and roast just until the feta begins to brown, 6 to 8 minutes.