Combine 1 large egg, 4 garlic cloves, finely grated, 1 cup panko, 2 Tbsp. extra-virgin olive oil, 1 Tbsp. thyme leaves, 1½ tsp. Diamond Crystal or 1 tsp. Morton kosher salt, and ¼ cup water in a medium bowl; season with freshly ground pepper and mix well. Let sit 5 minutes to hydrate.
Add 1 lb. ground beef to bowl and gently mix until combined. Divide and roll meat mixture into ping-pong-size balls (you should have about 20 total).
Heat remaining 2 Tbsp. extra-virgin olive oil over medium-high in a Dutch oven or other heavy, large pot. Cook meatballs, shaking pot to turn meatballs every few minutes, until browned all over, 6–8 minutes. Using tongs or a slotted spoon, transfer meatballs to a plate.
Cook 1 (packed) cup frozen, thawed pearl onions and ½ tsp. Diamond Crystal or ¼ tsp. Morton kosher salt in same pot, stirring occasionally, until slightly softened, about 2 minutes. Add 10 oz. mushrooms (any kind), torn into bite-size pieces, and cook, stirring occasionally, until browned, about 5 minutes. Add 2 Tbsp. double-concentrated tomato paste and cook, stirring often, until mushrooms are coated and paste is slightly darkened in color, about 1 minute. Evenly sprinkle 1 Tbsp. all-purpose flour over and cook, stirring constantly, until mushrooms are coated, about 1 minute.
Pour ²⁄₃ cup dry red wine into pot and cook, scraping up browned bits stuck to bottom of pot, until almost completely evaporated, 1–2 minutes. Stir in 1 lb. Yukon Gold potatoes (about 4 medium), peeled, cut into 1″ pieces, then add meatballs, 4 cups low-sodium beef broth, and remaining 1 tsp. Diamond Crystal or ½ tsp. Morton kosher salt; season with pepper. Increase heat to high and bring to a boil. Reduce heat, cover pot, and simmer soup, stirring occasionally, until potatoes are fork-tender, 15–20 minutes.
Remove lid and stir in ½ cup frozen peas; continue to simmer until liquid is reduced just slightly, about 5 minutes.
Ladle soup into bowls and serve with country-style bread if desired.
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