Remove the frozen cheesecake from the packaging and cut it into pieces. The pieces should be approximate squares. If the crust is crumbling away from the cheesecake, it may be best to trim the crust off the pieces as well. Keep the cheesecake pieces frozen while preparing the rest of the recipe.
Fill a deep skillet or Dutch oven with several inches of vegetable oil. Use a candy thermometer in the oil to monitor the temperature. Heat the oil until it reaches 375°F and maintain the oil temperature between 350°F and 375°F throughout cooking.
While the oil is heating, prepare the batter by whisking together the Bisquick pancake mix, milk, and one egg until well combined. The batter should be thick enough to coat the cheesecake pieces while still allowing you to drip off the excess and work quickly.
Dip one piece of cheesecake at a time into the batter. Use a fork to toss the cheesecake in the batter, and then lift the piece out and allow the excess batter to drip off. Immediately transfer the battered cheesecake into the hot oil.
Flip the cheesecake occasionally while frying to ensure even cooking. Remove the cheesecake from the oil when it is golden brown all over, about 1 to 2 minutes.
Blot excess oil from the fried cheesecake by placing it on several layers of paper towels.
Continue battering and frying all of the cheesecake pieces. Dust cheesecake pieces with powdered sugar.
Drizzle the sauces over the cheesecake bites or serve them alongside as dipping sauces.