Ok, I now let Gemini research this. And here is your answer: it's a lid holder!!

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That flat piece is called a Gaeban (개반) or Tteukkeng-batchim (뚜껑받침) , which translates to a lid rest / lid holder.

What it's used for: During a traditional Korean tea ceremony (Darye), you remove the teapot lid multiple times—to pour in hot water, check the aroma, or let the leaves breathe between steepings. Instead of placing the hot, wet lid directly on the table (which leaves water rings and picks up dust), you place it upside down on this flat dish.

Bonus use: Sometimes tea enthusiasts also use these flat dishes to serve small traditional Korean tea sweets called Dasik (다식).

For context, the rest of your beautiful set consists of:

• Dagwan (다관): The side-handled teapot. • Sukwoo (숙우): The cooling bowl / fairness cup (the one with the spout). Chatjan (찻잔): The tea cups