Preheat your oven to 375°F (190°C). Lightly grease a baking sheet or line it with parchment paper.
In a large mixing bowl, combine the all-purpose flour and butter. Use fingertips or two knives until mixture resembles coarse breadcrumbs (this is your dry mix for sopes). Add salt to taste if desired. Pulse ground chicken breast in a food processor with Monterey Jack cheese blend, beaten eggs and pepper. Mix thoroughly but do not overmix as it will make the mixture tougher when baked or fried.
Use an ice cream scoop to spoon out small rounds onto your prepared baking sheet. Bake for about 15 minutes, until golden brown on one side. Flip and continue cooking another 7-8 minutes more (or longer if needed). Keep warm after removing from the oven.
While sopes are still hot, spoon heap of cheese into each indentation. Allow to sit for a few moments until cheese has melted. Serve immediately with your favorite salsa on top.