Heat oven to 350 degrees. Place bread slices on two large rimmed baking sheets; brush with oil, and season with salt and pepper. Bake until golden, 15 to 20 minutes, rotating sheets halfway. Cool on baking sheets.
Reduce oven to 325 degrees. Heat canola oil in 5-quart dutch oven over medium-high heat. Season ribs with salt and pepper; brown on all sides, 12 to 15 minutes. Remove from pan; pour out all but 1 tablespoon oil. Add carrot, onion, and garlic; saute until lightly browned, 3 to 5 minutes. Add tomato paste, chipotles, and adobo sauce. Saute until caramelized, stirring often, about 2 minutes. Add wine and bring to a boil; add stock, and bring to a boil again.
Return ribs to pan, and cover; braise in the oven 4 hours. Remove ribs; break meat into 1-inch pieces and place in a bowl. Skim fat from braising liquid; puree vegetables with skimmed cooking liquid. Pour pureed sauce over meat; stir to coat. To assemble, lay an avocado slice on each piece of toast, top with a warm piece of short rib, and garnish with a serrano chile slice and a bit of cheese.