Step 1: Preheat oven to 350.
Step 2: Line a 9 inch square metal cake pan with aluminum foil, leaving 1 inch overhang for handles. Set aside.
Step 3: In a bowl whisk together oats, flour, brown sugar, and salt.
Step 4: Stir in melted butter and oil, until mixture clumps and is crumbly. Add another tablespoon of oil or water, if necessary, to ensure no dry spots.
Step 5: Add cheese, tossing to mix. Save 1 1/2 cups of mixture for later, and press the rest evenly into prepared pan.
Step 6: Bake in the center of a 350 degree oven, until edges are golden, about 15 minutes. Let cool.
Step 7: Peel, core, and slice apples 1/2 inch thick.
Step 8: In a saucepan, bring apples, sugar, lemon juice, and cinnamon to boil, adding up to 1/4 cup water, if apples stick to pan. If apples becomes too runny, sprinkle 1 teaspoon cornstarch over apples, then mix in, and let thicken. Reduce heat and simmer, stirring occasionally, until tender-crisp, about 5 minutes.
Step 9: Spread apple mixture over base. Sprinkle with remaining oat mixture.
Step 10: Bake until golden, about 30 minutes.
Step 11: Let cool in pan on rack. Cut into 3 inch squares.
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