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Preheat oven to 400 degrees F. In food processer fitted with S-shaped blade, combine cassava flour, almond flour, arrowroot, salt and thyme. Add butter and pulse until it forms pea-size pieces.
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Transfer mixture to bowl and set aside. In small bowl, combine water and egg yolk. Pour into flour mixture, stirring with fork until fully incorporated. Form dough into 6-in. disc, wrap and refrigerate 30 minutes.
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To prepare filling, warm medium sauté pan over medium heat. Add oil, onion and garlic and sauté 3-4 minutes until onions soften.
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Add mushrooms, salt and thyme and sauté 2 minutes. Add Brussels sprouts and sauté 2-3 minutes more.
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Transfer filling to bowl, stir in baby tomatoes and set filling aside to cool slightly.
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Roll out crust between two layers of parchment paper. Sprinkle paper with a little cassava flour if dough is too sticky. Roll crust into a 12-in. round, (don’t worry if it’s not exact). Peel off top layer of parchment paper.
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Spoon filling onto middle of crust, leaving 2-in. rim around it. Fold crust over edge of filling, toward the middle, using hands to gently press together. Sprinkle crumbled feta cheese over center.
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In small bowl, mix egg and water and brush over crust. Slide galette with bottom piece of parchment paper onto sheet pan.
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Bake 35-40 minutes until bottom and top crusts are lightly browned. Serve warm or at room temperature.