Combine vinegar, 1/2 cup water, 1 Tbsp. of the sugar, mustard seeds, and salt in a small saucepan; bring to a simmer. Pour vinegar mixture over carrot and onions in a clean, heatproof jar or bowl. Cover and chill at least 1 hour. Drain before serving.
Prepare Yum Yum Sauce by combining mayonnaise, coconut oil, tomato paste, garlic, paprika, and remaining 1/2 tsp. sugar. Whisk in 1 Tbsp. water to thin out to desired consistency. Set aside.
Heat oil in a skillet over medium heat. Add mushrooms and cook, tossing occasionally, until tender and golden brown, about 8 minutes. Add soy sauce, toss to coat and cook 1 minute longer. Remove from heat.
To assemble bowls, divide brown rice evenly into each of four bowls. Top with 1 Dr. Praeger's Korean Veggie Burger, sliced into quarters. Serve with pickled carrot slaw, shiitake mushrooms, and sliced avocado.