In a large bowl, toss beef with 3 tbsp. flour, 1 tsp. salt, and pepper. In a large pot over medium-high heat, heat 1 tbsp. vegetable oil. Add enough beef to pot to form a single layer (about 1/3 of beef), being careful not to overcrowd pot. Brown meat on all sides, about 7 minutes total per batch. Transfer to a large bowl and brown remaining beef in 2 batches. If meat or pan juices start to scorch, reduce heat.
Add 1 bottle ale to pot, scraping up brown bits with a wooden spoon. Pour ale over reserved beef and return empty pot to medium-high heat.
Add remaining tbsp. oil to pot. Add onions and 2 tsp. salt. Cook, stirring often, 2 minutes. Cover pot and reduce heat to low. Cook, stirring occasionally, until onions have a hint of gold color, about 20 minutes. Remove lid, raise heat to medium-high, stir in 3 tbsp. flour, and cook, stirring often, 3 minutes. Add mushrooms, reserved beef and ale, remaining bottle of ale, and carrots. Bring to a boil. Cover and lower heat to a gentle simmer. Cook, stirring occasionally, until beef is tender, about 3 hours.
About 40 minutes before serving, make dumplings: In a medium bowl, stir together remaining 2 cups flour, green onions, remaining 3/4 tsp. salt, and baking soda. Using a pastry blender or your fingers, work cold butter into flour mixture until it resembles cornmeal with some pea-size pieces. In another bowl, whisk together buttermilk and egg. Gently fold wet ingredients into dry, mixing until a very shaggy dough forms. If more liquid is needed, add additional buttermilk 1 tbsp. at a time. Gently form dough into 12 equal balls and drop into stew. Cover pot and cook 20 to 30 minutes, or until dumplings are fluffy and cooked through. Let sit 15 minutes before serving; stew will thicken as it cools.
Note: Nutritional analysis is per serving.
* Percent Daily Values are based on a 2,000 calorie diet. Your daily value may be higher or lower depending on your calorie needs.