Pulse bread cubes in food processor until small crumbles form. Spread them out evenly on a rimmed sheet pan and bake for 10 minutes. Remove, stir and bake for 5 more minutes. Remove and toss one last time before baking for 3 to 5 minutes. Allow breadcrumbs to cool in pan for 15 minutes.
Meanwhile, drain fresh mozzarella and blot between pieces of paper towel. Crack two eggs in a shallow bowl and beat with a fork.
Pour cooled breadcrumbs back into the food processor, add in 3/4 teaspoon kosher salt and pulse until fine crumbs form. You will have about 1 cup crumbs. Remove and pour into another shallow bowl.
Working in batches, toss fresh mozzarella balls in egg and drip excess off into dish. Toss in breadcrumbs and place on a clean platter. Repeat with remaining mozzarella.
Heat 1/4 inch of safflower oil in a deep-sided pan over medium-high heat. Once hot add a few of the breaded mozzarella balls and fry for 15 to 20 seconds before flipping and repeating. Transfer to a paper towel lined platter and repeat. Adjust the heat under the oil so the mozzarella bites do not burn.