Mix spring onions, flour, sun-dried tomatoes, sugar, baking powder, baking soda, and salt together in a large bowl.
Beat buttermilk, milk, melted butter, and egg together in a separate bowl until smooth.
Heat a lightly oiled griddle over medium-high heat.
Pour buttermilk mixture over flour mixture; mix together with a wooden spoon or fork until batter is just blended.
Pour 1/2 cup of the batter onto the hot griddle. Cook until bubbles form and the edges are dry, 3 to 4 minutes. Flip and top with some pepper Jack cheese and 1 slice of chicken. Cook until browned on the bottom, 2 to 3 minutes. Flip again and cook until chicken is warmed through, about 30 seconds. Repeat with remaining batter.