Heat olive oil in a large pot over medium-high heat until it shimmers slightly. Add your beef stew meat in a single layer and let it sit for three to four minutes without stirring—you want that nice brown crust. Flip each piece and cook another three minutes until browned on both sides. Don't panic if some pieces stick a little; that's actually the good stuff.
Push the beef to the side and add your chopped onion to the pot. Cook for two minutes, stirring occasionally, until the onion starts turning translucent. Add minced garlic and stir constantly for about 30 seconds until it gets fragrant. This whole step takes maybe three minutes, and it's worth every second.
Pour in your beef broth and use a wooden spoon to scrape up any brown bits stuck to the bottom of the pot (that's pure flavor). Add the diced tomatoes and stir everything together. Bring it to a gentle boil, then reduce heat to low and add your dried thyme and rosemary. Let it simmer for about 30 minutes so the beef starts getting tender.
After 30 minutes, add your sliced carrots and cubed potatoes to the pot. Give everything a good stir and add your salt. The broth should cover most of the vegetables; if it doesn't, add a little water. Let it keep simmering on low heat for about 35 to 40 minutes until the potatoes are fork-tender.
Taste your vegetable beef soup recipe and adjust the seasoning as you go. Sometimes mine needs an extra half-teaspoon of salt or a bit more thyme—every stove cooks differently. If it tastes a little flat, try adding a splash of Worcestershire sauce (seriously, game-changer move).
Once the potatoes are completely soft and the beef is tender enough to break apart easily, your soup is done. The whole cooking time is about 1 hour and 15 minutes from start to finish. Mine always looks slightly thicker the next day because the potatoes keep absorbing liquid, and that's completely normal.
Ladle into bowls and serve hot. I like to let it cool for about two minutes before eating since it gets super hot. If you're prepping ahead, let it cool completely before transferring to storage containers.