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Make the pasta dough: In a small saucepan, bring 1/2 cup water and saffron to a boil. Cook until reduced by half. Strain; let cool.
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Whisk together egg yolks, eggs, and saffron reduction in a medium bowl until well combined; set aside.
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In the bowl of a food processor, place flour, salt, and olive oil; pulse to combine. With the motor running, slowly add egg mixture and pulse until the dough resembles coarse breadcrumbs (you may not need to use all the egg mixture).
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Transfer dough to a lightly floured work surface, and knead until smooth, about 5 minutes. Wrap dough in plastic wrap, and let rest 2 hours before using.
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Make the filling: Heat oil in a large skillet over medium-high heat. Add mushrooms; season with salt. Cook until golden brown. Add arugula and cook until just wilted. Transfer mixture to a paper towel-lined plate to cool. Transfer to a medium bowl and stir in pine nuts and chestnut puree; set aside.
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To roll out the pasta dough, set the rollers of a hand-cranked or electric pasta machine at their widest opening. Dust dough lightly with flour, and run it once through the machine. Fold the dough in half, pressing it down with your fingertips, and run through the machine again. Repeat this step four or five times, dusting lightly with flour if needed, until the dough is smooth and elastic.
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Now change the rollers of the pasta machine to a decreasing setting, and roll out the dough once without folding. Keep rolling the sheet through the machine on decreasing settings until you have rolled it through the last (thinnest) setting.
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Flour a baking sheet; set aside. Cut dough into 12 rounds (3 1/2 inches each) and place 1 tablespoon of filling in the center of each round. Fold the dough in half over the filling to form a half-moon. Seal the dough around the filling, pressing the edges firmly together to seal. Bring the two pointed corners together around your finger and press to seal. Transfer tortellini to prepared baking sheet. Place in a single layer, leaving some space between them. Set aside, uncovered in the refrigerator until ready to use. Save any remaining pasta dough for another use.
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Make the soup: In a large saucepan over medium heat, add the artichokes, milk, heavy cream, and salt. Cook until artichokes are tender, about 15 minutes.
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Transfer mixture to a food processor and process until smooth. Return to saucepan and add chicken stock. Cook over medium-high until heated through.
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Bring a large pot of water to a boil. Add salt and 2 tablespoons olive oil and return to a boil. Add the tortellini and cook for 30 seconds; drain pasta and place 3 tortellini in the center of each of 4 serving bowls. Just before serving, stir butter into soup until incorporated. Pour over tortellini and serve immediately.
