Put the brisket in a large pot and fill with enough water to cover the meat by an inch. Add the onion quarters and 1/4 cup salt. Bring to a boil. Reduce the heat to medium and simmer partially covered until the brisket is very tender, 2 1/2 hours. Add more water if needed to keep the meat covered. Turn off the heat and let the brisket cool in the cooking liquid to room temperature. Drain the brisket, discard the water and cool completely.
Meanwhile, prepare the vinaigrette by whisking the olive oil and vinegar in a medium bowl until well combined. Add the lime juice, red onion and oregano and whisk again. Season the vinaigrette with salt and pepper.
Shred the brisket into a large bowl. Add the cucumbers, tomatoes, capers, cilantro and radishes and toss to combine. Add the vinaigrette and toss to coat. Season with additional salt and pepper if needed.
To make the grilled tostadas: Heat a grill pan on the stovetop over medium-high heat. Place a corn tortilla onto the hot pan until the tortilla is crisp with a slight char. Cook for about 5 minutes each side. Scoop the shredded beef salad onto the grilled tostada and garnish with some avocado slices.