Put 1/2 cup oats in a small bowl. Cover with 1/2 cup boiling water. Let stand until water is fully absorbed, about 5 minutes.
Coarsely chop the shallot and 1 carrot. Put in a food processor, along with the chickpeas. Add 1 tablespoon garam masala if you're using the chutney, 1 tablespoon barbecue spice if you're using the barbecue sauce, or 1 tablespoon chili powder if you're using the salsa verde; the soaked oats; the remaining 1/2 cup uncooked oats; and 1 teaspoon salt. Pulse, scraping the bowl occasionally, until mixture forms large clumps and vegetables are in small bits. Form into twelve 2 1/2-inch-diameter patties. Let stand 5 minutes.
Whisk the chutney, barbecue sauce, or salsa with the vinegar in a large bowl and add the cabbage. Grate the remaining carrot and add to the mixture. Add the remaining 1/2 teaspoon garam masala to the chutney, the remaining 1/2 teaspoon barbecue spice to the barbecue sauce, or the remaining 1/2 teaspoon chili powder to the salsa verde. Toss until evenly coated. Season to taste with salt.
Heat the oil in a large nonstick skillet over medium heat. Add the patties in a single layer. Cook, turning once, until browned on the outside and hot inside, 2 to 4 minutes per side. Transfer to plates.
Add the slaw and remaining sauce to the plates and serve.
Use one-inch-thick squares of Asian-style toast, like Japanese milk toast. You can buy it at Chinese, Japanese, and Korean bakeries.
Recipe by Genevieve Ko.