Instructions

Whisk flour, mushrooms, thyme, yeast, and salt together in bowl of stand mixer. Whisk whole egg and yolk, half-and-half, melted butter, and sugar in 4-cup liquid measuring cup until sugar has dissolved.
Fit mixer with dough hook and slowly add half-and-half mixture to flour mixture, mixing on low speed until cohesive dough starts to form and no dry flour remains, about 2 minutes, scraping down dough hook and sides of bowl as needed. Increase speed to medium-low and knead until dough is smooth and elastic and clears sides of bowl but sticks to bottom, about 8 minutes.
Transfer dough to lightly floured counter and knead by hand to form smooth, round ball, about 30 seconds. Transfer dough seam side down to lightly greased large bowl; cover with plastic wrap; and let rise until doubled in size, 1 to 1½ hours.
Line rimmed baking sheet with parchment paper. Press down on dough to deflate, then transfer to lightly floured counter. Press and roll dough into 15-inch circle. Brush top of dough with 1 tablespoon truffle oil and cut into 12 wedges. Working with 1 wedge at a time and starting at wide end, gently roll up each dough wedge, ending with pointed tip on bottom. Push ends toward each other to form crescent shape; transfer to prepared sheet with tip of dough underneath each roll, spacing rolls about 2 inches apart.
Cover sheet loosely with greased plastic and let rise until nearly doubled in size and dough springs back minimally when poked gently with your knuckle, 1 to 1½ hours. (Unrisen rolls can be refrigerated for at least 8 hours or up to 16 hours; let rolls sit at room temperature for 1 hour before baking.)
Adjust oven rack to middle position and heat oven to 350 degrees. Gently brush rolls with egg wash and bake until golden brown, 20 to 25 minutes, rotating sheet halfway through baking. Transfer rolls to wire rack, brush with remaining 2 teaspoons truffle oil, and let cool for 15 minutes. Serve warm.