01 - Preheat the oven to 425°F. Line a baking sheet with parchment paper.
02 - In a large bowl, whisk together flour, baking powder, baking soda, sugar, and salt. Use a pastry blender or fingertips to cut in cold butter until the mixture resembles coarse crumbs.
03 - Pour in cold buttermilk and stir gently until just combined. Turn the dough onto a floured surface and pat into a 1-inch-thick rectangle. Fold the dough into thirds, rotate, and repeat this process twice to create flaky layers.
04 - Cut out 4 large rounds with a 3-inch biscuit cutter and arrange them on the lined baking sheet. Brush biscuit tops with extra buttermilk. Bake for 14 to 16 minutes, until golden brown. Let cool slightly.
05 - Pat chicken breasts dry. Sprinkle both sides evenly with Cajun seasoning. Heat olive oil in a skillet over medium-high heat. Cook chicken for 5 to 6 minutes per side until golden brown and cooked through, reaching an internal temperature of 165°F. Rest chicken for 5 minutes before slicing thinly.
06 - Whisk eggs with milk, salt, and pepper in a bowl. Melt butter in a nonstick skillet over medium heat. Add the egg mixture and gently scramble until just set.
07 - Split warm biscuits in half. Layer with scrambled eggs, sliced Cajun chicken, cheese slices, and green onions if desired. Replace the biscuit lids and serve immediately.