Use any smoked cheese or other greens, such as spinach or kale. You can serve this with warmed chunky marinara sauce.
Preheat oven to 450°.
Combine boiling water and 1/3 cup cornmeal; let stand 20 minutes, stirring occasionally. Dissolve yeast in warm water; let stand 5 minutes.
Lightly spoon flour into dry measuring cups; level with a knife. Combine cornmeal mixture, flour, 1/2 teaspoon salt, and thyme in a food processor; pulse 4 times or until blended. With processor on, slowly add yeast mixture, honey, and oil through food chute; process until dough forms a ball. Process 1 additional minute. Turn dough out onto a floured surface; knead lightly 4 or 5 times (dough will feel tacky).
Place dough in a bowl coated with cooking spray, turning to coat top. Cover and let rise in a warm place (85°), free from drafts, 1 hour or until doubled in size. (Gently press two fingers into dough. If indentation remains, dough has risen enough.)
While dough is rising, remove stems from greens. Wash and coarsely chop. Bring 4 cups water to a boil in a Dutch oven. Add greens, and cook 1 minute, stirring constantly. Remove greens from water with a slotted spoon. Plunge into ice water, and drain. Squeeze out any remaining liquid.
Heat pan coated with cooking spray over medium-high heat. Add mushrooms and garlic; sauté 5 minutes or until just tender. Add greens, 1/2 teaspoon salt, vinegar, black pepper, and red pepper; cook 1 minute.
Punch dough down. Cover and let rest 5 minutes. Divide dough into 4 equal portions. Working with one portion at a time (cover remaining dough to prevent drying), shape each portion into a ball. Roll out into a 7-inch circle. Place on a baking sheet lightly dusted with 2 teaspoons cornmeal.
Place 1/2 cup greens mixture on half of each circle leaving a 1-inch border; sprinkle each circle with 1/4 cup cheese. Fold dough over filling until edges almost meet. Bring bottom edge over top edge, and crimp the edges with fingers to form a rim. Place calzones on a baking sheet coated with cooking spray. Combine 1 1/2 teaspoons water and egg white, stirring with a whisk. Brush the calzone tops with egg mixture. Bake the calzones at 450º for 12 minutes or until golden brown.