To make the dough, in a large mixing bowl, combine flour, yeast and salt. Whisk together.
Add vegetable oil and water and mix to combine ingredients using a wooden spoon or dough whisk. Knead until dough is smooth and elastic. (If dough is too dry, add more water a tablespoon at a time until dough is smooth and soft. If the dough is too wet, add more flour, a tablespoon at a time until dough is smooth and soft.)
Coat the dough with a little bit of vegetable oil and place it back in the bowl. Cover with damp kitchen towel and place it in a warm place to rise, for about 30-60 minutes.
Prepare the filling. Heat olive oil in a large skillet or Dutch oven over medium high heat. Add the meat, garlic and salt. Cook the meat, breaking it apart with a wooden spoon, until cooked through. Transfer into a large bowl and mix it with rice.
Once the dough is doubled in size, take it out onto a lightly floured surface and knead into a smooth ball, 1-2 minutes. Lightly flatten the dough ball into a disk, and divide into 12 equal parts as you slice a pizza. Roll each piece into a smooth ball.
Make the hand pies: take a piece of dough and flatten it with your hands. Put the filling in the center and fold the dough in half. Pinch the edges of the dough to seal, creating a half moon. Then gently flatten it between your palms, making sure the sealed crimps are on flat side.
Once all the hand pies are formed, heat 1-inch oil in a large skillet, or Dutch oven. (Tip: You don't need a thermometer to fry these savory hand pies. Simply drop the end of a wooden spoon into heated oil, and if it's bubbling up around the wood, that means the oil is ready for frying!)
To cook the hand pies, drop 4 hand pies in the hot oil and cook until golden brown, 3-4 minutes. Flip and cook until it's nice and golden. Remove into a bowl, lined with paper towel to absorb excess oil. Continue with the remaining batches.
Serve piroshki warm or at room temperature.