In a bowl, combine water, yeast and sugar. Mix to combine and let sit in a warm place until frothy.
In a mixing bowl with dough hook attachment combine flour, 1 tablespoon of salt, 1/2 cup of olive oil, and yeast mixture. Mix on low speed until dough comes together. Increase to medium speed and knead 5 to 6 minutes more adding flour if dough is too sticky. Remove and knead on tabletop until smooth. Place dough into a bowl and cover well with foodservice film. Allow to rise in the refrigerator overnight.
Coat a 1/2 a sheet pan with the remaining olive oil. Press dough into oil until it fills the pan. Allow to rise until doubled. Preheat oven to 425ºF. (What is an "a" sheet pan?)
In a small bowl combine remaining sea salt, 1/2 teaspoon ground jalapeno pepper, and lime zest. Sprinkle over focaccia and bake for 25 minutes. Remove and let cool.
In a food processor, combine 3 halves of avocado (peeled and pitted), lime juice, sea salt, and 1/4 teaspoon ground jalapeño pepper. Puree until smooth. Coarsely mash the last half of avocado in a bowl. Add the puree and mix well to combine. Spread on pieces of focaccia, dust with pepper or ground jalapeño and serve.