Heat a medium, heavy pot over medium-high heat 2 minutes. Swirl in oil to coat bottom. Add beef in a single layer, working in 2 batches if necessary, and brown well all over, about 10 minutes.
Reduce heat to medium-low. Add onion and apple and cook, stirring often, until onion is very soft and golden brown, 15 to 20 minutes. Add garlic, ginger, sake, mirin, and 1/2 tsp. salt and cook, stirring, until liquid is almost evaporated, 3 to 5 minutes.
Add curry powder or chopped block curry, garam masala, soy sauce, brown sugar, and chicken broth and simmer, uncovered and stirring occasionally, until beef is tender, about 30 minutes. Add carrot, potato, and squash and cook until vegetables are tender, about 20 minutes. Season with salt to taste.
Meanwhile, boil udon (see "Udon Essentials," below). Using a large strainer, scoop out noodles into a large bowl and save water to heat soup bowls.
Mix cornstarch, flour, and 2 tbsp. water in a small bowl until smooth. (Omit this step if you are using chopped block curry.) Set a fine-mesh metal strainer in pot of curry and slowly pour in cornstarch mixture, stirring within the strainer (this will prevent lumps). Remove strainer and continue simmering, stirring often, until curry thickens, about 5 minutes.
To serve, warm 4 soup bowls by dipping them into hot udon-cooking water. Divide noodles among bowls and top with curry.
Udon Essentials Udon (wheat-flour noodles): Store-bought fresh-frozen noodles have a supple texture that's closest to homemade, while the dried ones tend to be thin and flabby. To cook store-bought fresh-frozen udon, drop the frozen block into boiling water. When the water boils again, drain. Cook udon right before serving; the noodles get sticky as they sit.
*Find garam masala, an Indian spice blend, in the spice aisle of well-stocked grocery stores and at Indian markets.
Make ahead: Curry, up to 4 days, chilled.
* Percent Daily Values are based on a 2,000 calorie diet. Your daily value may be higher or lower depending on your calorie needs.