This baked potato cakes are every bit as flavorful as their all-fried counterparts.
Place potatoes in small stockpot, cover with water, and bring to a boil. Reduce heat to medium high; gently boil until tender, 40 to 45 minutes. Drain in a colander. Using a towel to hold potatoes, peel while still hot. Pass potatoes through a potato ricer or a food mill into a large bowl.
Heat oven to 450 degrees. Meanwhile, heat olive oil in skillet over medium-low heat. Add garlic, onions, red peppers, and celery; saute until vegetables are translucent, about 5 minutes. Add stock; saute 1 minute more. Add vegetable mixture to potatoes along with milk, thyme, salt, and black and cayenne peppers. Form 1/3 cup mixture into 3-inch round cake; transfer to platter. Repeat, making cakes with remaining mixture.
In a shallow dish, whisk egg whites until foamy. Place bread crumbs in a shallow dish. Coat cakes in egg whites, then in bread crumbs; transfer to a baking sheet.
Pour 1 tablespoon canola oil in a medium nonstick skillet; place over medium-high heat. Add half the cakes; cook until golden brown, 2 to 3 minutes on each side. Return to baking sheet; set aside. Repeat process with remaining tablespoon canola oil and cakes, wiping skillet with paper towel between batches.
Place cakes in oven; cook until deeply golden, about 20 minutes. Serve with Sweet Red-Pepper Sauce.