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1
Core apples and cut into 1/2-inch dice. Cut leeks in half lengthwise, then cut into 1/4-inch half moons. (Make sure to clean your leeks well, dirt likes to hide between the layers!) Core and roughly dice fennel bulb.
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2
Adjust oven rack to lower-middle position and heat oven to 300 degrees. Pat beef dry with paper towels and season with salt and pepper. Heat 1 tablespoon oil in large heavy-bottomed Dutch oven over medium-high heat until smoking. Add half of beef and cook on each side until well browned. Do not move the pieces until nice crust has formed, which should take about 4 to 5 minutes per side. Transfer beef to medium bowl. Repeat with remaining tablespoon oil and ribs.
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3
Pour out all but 2 Tablespoons oil. Reduce heat to medium, add apples, leeks, and fennel (or celery), and cook, stirring occasionally, until softened and just starting to brown (about 10 - 15 minutes.) While they are cooking mince the garlic. Add garlic and caraway seed (and fennel seed if using) cook until fragrant, about 30 seconds. Add flour and stir to combine for about 1 minute. Add broth, beer, espresso, mustard, thyme, and bay leaves, then stir, scraping bottom of pan to loosen any stuck bits. Add ribs and any accumulated juices to pot, nestling the ribs in, so they are submerged. Bring to a simmer over medium-high heat. Cover and transfer pot to oven and cook until tender, turning ribs once at about 90 minutes. Ribs will take at least 2 & 1/2 to 3 hours to cook. Transfer pot to wire rack and allow to cool partially covered for about 1 hour.
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4
Transfer ribs to a large plate, scraping off excess apple and veggie bits clinging to the ribs. Strain liquid into medium bowl, pressing the solids to extract as much liquid as possible; discard solids. Transfer liquid into a fat separator, allow to sit for 5 or so minutes, then pour liquid into clean bowl. Discard the fat. You may need to do this in batches. Clean your dutch oven.
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5
Melt 2 tablespoons butter in dutch oven over medium-low heat. Once it has stopped sputtering, add the sliced shallots and a pinch of brown sugar. Cook, stirring occasionally until the shallots are lightly caramelized.
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6
Add the braising liquid to the dutch oven, and cook over medium-high heat until reduced by about 1/3 to 1/2 depending on personal preference. Taste the liquid, and add salt and pepper to taste. Add brown sugar to taste (1 to 2 tablespoons) to temper bitterness from the beer and caraway. Reduce heat to medium, and add the ribs back to the pan. Simmer until the ribs are heated through, stirring and flipping ribs occasionally, about 8 to 10 minutes. Serve with something that can soak up the sauce.
