Place the peeled and chunked potatoes into a large pot and cover with cold water. Add 1/2 teaspoon salt to the water.
Bring the pot to a boil over high heat, then reduce heat to medium and simmer until potatoes are tender, about 15-20 minutes.
While the potatoes cook, heat olive oil in a large skillet over medium-high heat.
Add the chopped onion and cook for 3-4 minutes until softened.
Add the minced garlic and cook for 1 minute until fragrant.
Add the ground beef to the skillet, breaking it apart with a spatula. Cook until browned and no longer pink, about 7-8 minutes.
Season the beef with 1 teaspoon salt, 1/2 teaspoon black pepper, and 1/2 teaspoon dried thyme. Stir to combine.
Pour in the beef broth, reduce heat to medium-low, and simmer for 5 minutes until slightly thickened.
Drain the cooked potatoes and return them to the pot.
Add the butter, milk, 1/2 teaspoon salt, and 1/4 teaspoon black pepper to the potatoes.
Mash the potatoes with a potato masher or fork until smooth and creamy.
Serve the ground beef mixture over or alongside the mashed potatoes.