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1. In a small bowl, mix together 1/4 cup brown sugar, 1/4 cup hoisin and 2 Tbsp Korean chili paste. Spread all over beef roast. Place roast into pressure cooker. Pour in 1 1/4 cup beef broth. Close lid and cook on high pressure for 15 minutes. Let pressure naturally release for 10 minutes and then, being careful to avoid the steam, manually release the pressure. Shred beef.
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2. In a saucepan, whisk together 2 Tbsp Korean chili paste, 1/3 cup ketchup, 1/4 cup rice vinegar, 2 Tbsp brown sugar, 2 Tbsp soy sauce, 4 Tbsp honey and 4 minced garlic cloves. Bring to a boil and then reduce heat to low. Simmer, whisking occasionally, for 10 minutes. Remove from heat and add 1/2 to shredded beef. Stir to combine.
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3. In a medium bowl, combine carrots, cucumber, onion and 1/4 cup rice vinegar. Season with salt and pepper. Refrigerate for at least 15 minutes.
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4. Spread remaining sauce on each of 12 Pepperidge Farm Slider Buns. Top with shredded beef and vegetables. Serve immediately.