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Combine 1 cup warm water, honey, and 1 teaspoon oil in a small bowl, stirring with a whisk. Stir in yeast; let stand 10 minutes. Weigh or lightly spoon flours into dry measuring cups; level with a knife. Place flours and salt in a food processor; pulse 2 times or until blended. Add yeast mixture, pulsing to combine (dough will feel sticky). Turn dough out onto a floured surface; knead lightly 3 to 4 times.
Place dough in a large 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.) Punch dough down; cover and let rest 10 minutes.
Preheat oven to 450°.
Place a baking sheet in oven. Roll dough into a 14-inch circle on a floured surface. Brush dough evenly with 1 tablespoon oil, and sprinkle evenly with cheese. Place dough on a baking sheet sprinkled with cornmeal. Transfer dough carefully to preheated pan; bake at 450° for 12 minutes or until crust is crisp and golden. Combine remaining 2 teaspoons oil, mâche, and next 5 ingredients (through peas); toss to combine. Arrange salad over crust.