Preheat oven to 375 °F.
Heat olive oil in pan over medium-high heat; add garlic. Cook garlic 2 minutes, and then add spinach. Cook until spinach is wilted, stirring frequently.
Remove from heat and stir in cheeses; set aside to cool.
Separate pieces of dough, and place roughly 2 tablespoons filling in middle of each dough circle. For each empanada, fold dough over on top of filling to close. Then, wet fingers with water and run them along edges to help seal the empanada. Be sure to press edges firmly together.
Transfer empanadas to baking sheet. Take beaten egg and brush across tops of empanadas. (This will give them shine when they bake.)
Place in oven, and bake for 20-25 minutes, or until golden brown.