Heat 1 tablespoon oil and 1 tablespoon butter in a large saute pan over medium heat. Add sliced garlic and sliced lemon and cook until golden brown, about 3 minutes. Add flour and stir to coat. Add red pepper flakes and white wine and bring to a boil. Continue to cook until almost all of the white wine has evaporated.
Add chicken stock and capers and bring to a boil. Reduce heat to low and simmer for 10 minutes or until sauce has reduced by half. Add remaining 2 tablespoons butter, whisking until butter is fully incorporated in sauce. Remove sauce from heat and add lemon juice. Season with salt and pepper and set aside.
Heat frying oil to 400 degrees F.
Whisk together club soba, rice flour, paprika, salt, and pepper until well combined. Dip sliced calamari in batter, tossing to coat.
Carefully drop calamari into hot oil and fry until golden brown, about 2 minutes. Remove calamari with slotted spoon and place on paper towels to drain excess oil. Repeat with remaining calmari.
Drizzle piccata sauce on calamari, tossing gently to coat. Transfer calamari to serving platter and sprinkle chopped parsley and grated parmesan on top. Serve immediately.