Bring pork, quartered onion, smashed garlic, thyme, 1 tsp salt and 6 cups water to simmer in large saucepan over medium-high heat, skimming off any foam that rises to the surface. Reduce heat to medium-low, partially cover and cook until pork is tender, 75-90 minutes. Drain pork, reserving 1 cup cooking liquid. Discard onion, garlic and thyme. Return pork to saucepan and using potato masher, mash until shredded into rough 1/2" pieces. Set aside.
Heat olive oil in 12" nonstick skillet over medium-high heat until shimmering. Add shredded pork, chopped onion and oregano; cook, stirring often until pork is well browned and crisp, 7-10 minutes. Add minced garlic and cook until fragrant, about 30 seconds.
Stir in tomato sauce, chipotle powder, reserved pork cooking liquid and bay leaves; simmer until almost all liquid has evaporated, 5-7 minutes. Remove and discard bay leaves and season to taste with salt.
To fry tostadas, heat vegetable oil in 8" heavy bottomed skillet over medium heat to 350 degrees. Using fork, poke center of each tortilla 3-4 times (to prevent puffing and allow for even cooking). Fry 1 at a time, holding tortilla down to keep it submerged. Drain on paper towel lined plate and season with salt to taste.
Spoon small amount of shredded pork onto center of each tostada and serve with garnishes.