Step 1In a large bowl, mix together the thyme, oregano and oil. Add the tomatoes and toss to combine. Heat a grill (or grill pan) over medium-high heat until hot. Grill the tomatoes until softened, about 4 to 5 minutes depending on the heat. Remove the tomatoes to a blender and pulse until smooth. Combine the softened tomatoes with the liquid smoke and mayonnaise in a large bowl and mix well to combine. Season to taste with salt. This makes about 2 cups aioli, more than is needed for the recipe; the aioli will keep, covered and refrigerated, for 1 week.
Step 1Heat a grill or grill pan over medium-high heat until hot. Brush with a thin coat of oil, then add the pineapple slices and grill until nicely charred, 2 to 4 minutes on each side. Cool the pineapple, then dice into one-fourth- to one-half-inch pieces. In a large saucepan, combine the pineapple, onion, jalapeno, vinegar, water, honey, sugar and salt over low heat, and cook until almost all of the liquid is evaporated, about 45 minutes, stirring occasionally. Chill, uncovered, on a rimmed baking sheet in the refrigerator. This makes 4 cups relish, more than is needed for the recipe; the relish will keep, covered and refrigerated, up to 2 weeks.
Step 1In a large saute pan, combine the onion, garlic, carrot, squash and zucchini with the wine over medium-high heat. Cook the mixture until the vegetables are softened and the liquid is mostly absorbed, 12 to 14 minutes.
Step 2Stir in the cumin, curry powder, salt, black pepper, oats and black beans. Continue to cook, stirring gently, for an additional 5 minutes to develop the flavors. Remove from heat and set aside to cool.
Step 3In a large bowl, whisk together the egg and soy sauce, then add the cheese and panko crumbs. Add the cooled vegetable mixture and stir well to combine.
Step 4Divide the mixture evenly into 6 mounds. Press each mound together, forming a patty. Lightly dust the patties with flour.
Step 5Heat a griddle or large saute pan over medium-high heat. Grease with a little oil, then add the patties (do not crowd; this may need to be done in batches). Pan-fry until crisp and dark brown on each side and the center is cooked through, about 5 minutes. Timing will vary depending on the heat of the griddle or pan, cover the griddle or pan with a lid so the center cooks along with the outside of the burgers. Keep the burgers warm until all are cooked through.
Step 6To assemble each burger: Spread the inside of each side of the bun with 2 tablespoons tomato aioli, or to taste. Place a burger on the bottom bun, then top with a sprinkling of piquillo peppers. Layer with sliced avocado, then top with 3 tablespoons pineapple relish, or to taste. Place top bun on each burger. Serve immediately.