Preheat the oven to 425 degrees F. Line a baking sheet with parchment paper. Combine the flour, baking powder, sugar and 1 teaspoon salt in a food processor and pulse until combined. Add the butter; pulse until it is in pea-size pieces. Add the cheddar and scallions and pulse 2 or 3 times to incorporate. Add the buttermilk and pulse a couple of times until the dough just comes together.
Turn out the dough onto a lightly floured surface. Pat into a 6-by-7-inch rectangle (1/2 to 3/4 inch thick). Cut out biscuits with a floured 2-inch biscuit cutter and transfer to the prepared baking sheet. Pat the scraps together and repeat to make 16 biscuits total. Brush the tops with buttermilk. Bake until lightly golden and puffed, 12 to 15 minutes. Transfer to a rack and let cool.
Meanwhile, working in batches, cook the bacon in a large skillet over medium heat, turning, until crisp, about 5 minutes. Drain on paper towels. Break each slice into thirds.
Mash the avocado in a small bowl with the lime juice and jalapeno brine; stir in the chopped jalapenos and season with salt. Season the tomatoes with salt.
Split the biscuits; spread the top halves with the mashed avocado. Fill with the bacon, lettuce and tomatoes.