In a large saucepan, cover the walnuts with water. Simmer over moderately low heat until soft, about 50 minutes; drain. Spread the nuts on a rack to dry.
In a medium saucepan, combine the sugar with the 1/2 cup of water. Bring to a boil and simmer over moderately high heat until the syrup reaches 230° on a candy thermometer, about 15 minutes. Add the walnuts and stir to coat them completely. Transfer the walnuts to a baking sheet in an even layer and let cool.
In a large saucepan, heat the oil until it reaches 350°. Fry the walnuts in 4 batches: for each batch, carefully lower the nuts into the hot oil and when it bubbles up, stir well. Fry the nuts until shiny and crisp, about 4 minutes. With a slotted spoon, transfer the nuts to a rack set over a baking sheet and sprinkle with some of the sesame seeds. Repeat with the remaining walnuts, reducing the heat if the oil gets too hot.
The nuts can be stored in an airtight container for up to 1 week.
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