Place the diced potatoes in a medium pot and cover with water. Bring to a boil over high heat and cook until tender, about 15 minutes.
While the potatoes cook, heat olive oil in a large skillet over medium heat.
Add the chopped onion and diced carrot to the skillet and cook until softened, about 5 minutes.
Add the minced garlic and cook for 1 minute until fragrant.
Add the ground beef to the skillet. Cook, breaking it up with a spoon, until browned and cooked through, about 7-8 minutes.
Season the beef mixture with salt, black pepper, dried oregano, and dried basil. Stir well to combine.
Add the frozen peas and beef broth to the skillet. Stir and simmer for 5 minutes until the liquid reduces slightly.
Drain the cooked potatoes and return them to the pot.
Add the milk and butter to the potatoes. Mash until smooth and creamy.
Stir in the shredded cheddar cheese into the mashed potatoes until melted and well combined.
Serve the seasoned ground beef and vegetable mixture alongside the cheesy mashed potatoes.