Heat the oil in a large skillet over medium-high heat. Add the carrot, spinach, and ground beef. Stir often, breaking up the meat into crumbles, until the meat is cooked through and the vegetables are soft.
Season with salt, garlic powder, onion powder, and pepper. Stir in the Worcestershire sauce, ketchup, and 1/4 cup water. Scrape up any brown bits stuck to the bottom of the pan and slowly simmer until most of the liquid is gone.
Transfer the meat mixture to a bowl and set aside to cool.
Heat the oven to 400 degrees and line a sheet pan with parchment paper or grease lightly with oil.
Open your can of biscuits and place them on the baking sheet. Carefully split a biscuit in half. Place the bottom half of the biscuit on the baking sheet and use your finger tips to gently press the dough into an even, slightly bigger circle.
Once the meat mixture has cooled (it can still be warm but not pipping hot), place a heaping spoonful in the center of the bottom biscuit dough then top with a big pinch of the shredded cheese.
Use your fingers to pinch the top part of the biscuit into an even, slightly bigger circle that will fit over the bottom biscuit and fillings. Place on the top dough and use your finger tips to press the edges of the dough together. Use the tines of a fork to seal the edges of the dough and then poke the fork through the top of the biscuit to create an air vent.
Repeat with the remaining dough and filling. Brush the top of the biscuits with the beaten egg then sprinkle with the sesame seeds, if using.
Bake the biscuits until golden brown, about 15 to 20 minutes. Serve warm or store in the fridge or freezer to reheat later (see notes).