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Filling and Sauce: Preheat oven to 350°. Season lamb with salt and pepper. Heat canola oil in a medium ovenproof pot over medium-high. Cook lamb, turning occasionally, until evenly browned, 10–12 minutes. Transfer to a plate.
Reduce heat to medium-low and cook shallots and garlic until shallots are softened and edges of garlic are golden brown, 3–4 minutes. Add tomato paste and cook until heated through and darkened in color, about 2 minutes. Add wine, scraping up browned bits, and bring to a simmer. Add stock, bay leaves, 2 thyme sprigs, and reserved lamb shank; return to a simmer. Cover, transfer to oven, and braise, turning lamb occasionally, until meat is falling off the bone, 2½–3 hours.
Meanwhile, tie remaining 5 thyme sprigs into a bundle with twine. Heat olive oil in a large skillet over medium. Add onions and thyme bundle; season with salt. Cook, stirring frequently, until onions are deep golden brown, 25–30 minutes. Remove thyme bundle and discard. Finely chop onion; set aside.
Transfer lamb to a cutting board and let cool. Strain braising liquid through a fine-mesh sieve into a small bowl or measuring cup (you should have about ¼ cup). Discard thyme and bay leaves; separately reserve liquid and solids. Wipe out pot.
Pull off meat from shank and finely chop; transfer to a medium bowl; discard any silver skin and excess fat. Add caramelized onions, cheese, 4 Tbsp. braising liquid, and half of the braising solids. Stir to combine; taste and adjust seasoning with salt and pepper.
Crush tomatoes with your hands or the back of a fork in reserved pot. Add remaining braising liquid and solids and cook over medium heat, stirring occasionally, until reduced by half and sauce is thickened and slightly chunky, 25–35 minutes. Stir in butter and vinegar; season with salt and pepper.
Assembly: Dust a large rimmed baking sheet with cornmeal. Set pasta maker to thickest setting; dust lightly with cornmeal. Divide dough into 4 pieces. Working with 1 piece at a time and keeping remaining dough wrapped in plastic as you work, flatten dough into a narrow rectangle (no wider than mouth of machine); pass through rollers. Fold dough as needed to fit and roll again. Repeat without folding, adjusting machine to thinner settings after every pass and dusting with cornmeal if sticky, until pasta is 1/16" thick (setting 8 on most machines). (Alternatively, you can roll out sheets lengthwise with a rolling pin until 1/16" thick.)
Lightly dust work surface with cornmeal. Cut sheet of dough in half crosswise. Starting 1" from the short edge, spoon lamb mixture, 1 Tbsp. at a time, down length, spacing about 1" apart (you should get 5 mounds). Brush dough and top of filling with egg wash. Top with second length of dough; press around each mound to seal, then press outward toward edges, pushing out any air pockets.
Using a sharp knife or pizza cutter, cut each ravioli into a 2" square. Arrange on prepared sheet, dust with cornmeal, and cover loosely with plastic wrap. Repeat with remaining dough and filling.
Working in batches if needed, cook ravioli in a large pot of boiling salted water until tender, about 5 minutes. Drain.
Divide ravioli among plates; cover with tomato sauce. Top with cheese and thyme.
Do Ahead: Assembled ravioli can be made 3 hours ahead; cover and chill.