Teriyaki Mushroom and Egg Noodle Bowls

Name: Teriyaki Mushroom and Egg Noodle Bowls
Ingredients: 1 tablespoon olive oil, plus more for brushing; 5 tablespoons salted butter; 1 pound assorted mushrooms, thickly sliced or quartered; 6 scallions, cut into 1-inch pieces; 2 tablespoons grated fresh ginger; 3 cloves garlic, finely chopped; 1/2 cup plus 1 tablespoon bottled teriyaki sauce; 2 bunches spinach, tough stems trimmed (about 1 pound); Soy sauce, to taste; 4 to 6 large eggs; Kosher salt; 4 cups cooked soba noodles; Toasted sesame seeds, for sprinkling