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Heat an InstaPot to high heat using the saute setting. Season short ribs on all sides generously with salt and pepper. Add olive oil to InstaPot and sear short ribs on all sides until browned, about 2-3 minutes per side. Remove from pot and set aside.
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Bring a large pot of salted water to boil and cook manicotti noodles until al dente. Drain and allow to cool to room temperature.
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Add leeks to InstaPot and cook until they becomes fragrant and softened, about 6-8 minutes. Add garlic and saute for about 1 more minute, stirring occasionally. Add tomato paste and stir for 30 seconds until everything is nicely coated, then deglaze the pot with red wine. Allow to cook for about 2-3 minutes until the wine has reduced by about half.
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Add stock, mustard, thyme, salt and pepper and stir to combine. Add beef short ribs back to the pot. Replace lid of InstaPot and set to Pressure Cook mode on high for 45 minutes.
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While short ribs are cooking, prepare ricotta filling and manicotti. Whip together fresh ricotta, heavy cream, lemon juice, lemon zest and salt in a mixing bowl until combined.
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Heat oven to 400 degrees Fahrenheit.
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When short ribs are finished cooking, allow to natural release the steam for 10 minutes before venting. Remove short ribs and discard bones. Shred meat and return to the pot, stirring to coat. If sauce remains thin, you can turn the saute function on again for 5-10 minutes to help thicken the sauce.
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Spray a large baking dish with nonstick cooking spray. Use a spoon or piping bag to fill each cooked manicotti noodle with ricotta filling, then arrange them in one even layer in the prepared baking dish. Spoon short rib sauce over the top and cook in oven for 25-30 minute. Enjoy!