Here, Eat This: A Beginner's Guide to Japanese Cuisine (That's Not Sushi or Ramen)
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Not to be confused with donburi, udon is a noodle soup (and a noodle soup that's not to be confused with ramen). Got it? In its most basic form, udon is simply a bowl of thick wheat flour noodles in a broth made with dashi, mirin and soy sauce. It's most commonly topped with fried meat or seafood, scallions and fish cakes called kamaboko. The extra thick, jiggly udon noodles are easily the most slurpable noodles you'll find in Japanese cuisine.
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Like udon, soba noodles are a popular alternative to rice. Unlike udon, soba noodles are made from buckwheat and can be served hot or cold. Chilled soba noodles are usually served separate from the tsuyu broth, which you use to dip your cold noodles into, as many fans of soba say that the texture is best when the noodles haven't been sitting in warm soup. Hot soba dishes are served in a thinner version of the tsuyu, and can be topped with anything from fried shrimp to raw eggs that cook in the broth.
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Gyoza go by many names from culture to culture; they're the dumplings that are also called jiaozi in China or pierogie in Poland or manti in Turkey. In Japan, gyoza are usually filled with pork, cabbage, garlic and ginger. They're pan fried to give one side a crispy texture, then steamed to finish. As with tempura and tonkatsu, gyoza were adapted from a foreign cuisine (Chinese, in this case) but are considered "Japanese" by now.
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Umeshu has an unusually sweet, floral fragrance and taste that comes from steeping the ume fruit (which is halfway between a plum and an apricot) in alcohol and sugar until fermented. The result is a liquer that's between 10 and 15 percent ABV, so this isn't something you drink throughout a meal. I think of umeshu (pronounced: woo-MAY-shoo) as a cousin to sherry or Port, and consume it in a similar fashion.
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Although you may have seen sake called "rice wine," it's truly more like "rice beer." Rice is fermented to make Japan's most popular alcoholic beverage, but there are far more factors at play than just that. The many varieties of sake are differentiated by the type of rice used, how long it was polished, how long the sake ferments, what water is used, how long the sake matures and how filtered the sake is before it's bottled. Sake can be filtered or unfiltered, dry or sweet, fruity or floral and can be served hot or cold. There are simply too many iterations on this fascinating beverage to list, so seek out a sake expert and dive in for yourself if you're interested -- it's rich, underexplored territory in the United States.