In a big, sturdy bowl, stir together the warm water and sugar
Sprinkle the packet of yeast over the warm water, and leave it to sit for 5 minutes or so
Add 3 tablespoons of the oil and 3 cups of the flour, stirring til well combined. Then add the salt and 1 1/2 cups of flour, stirring until there are no bog lumps or major dry spots.
Sprinkle some of the remaining 1/2 cup of flour onto your kneading surface and knead for 10 minutes, adding the rest of the flour as you work to keep your dough from sticking on your hands.
If you need a little more or less flour that is fine, just keep working until you can form a smooth ball of dough. Once your dough is ready it should be smooth, elastic, and noticeably less sticky than before. It's ok if it takes a little longer than 10 minutes.
Add a splash of oil back to the mixing bowl and return the dough to the bowl, flipping it once or twice to be sure it's well-coated in oil. Cover with a damp towel and a plate, leaving it in a warm place for 90 minutes. Take a break and have a snack, remember to hydrate!
After the hour is up the dough should be about doubled in size. Punch it down and fold it over once or twice, getting rid of any major bubbles. Transfer your dough to a lightly oiled 9"x13" rectangular pan, stretching and spreading it out with your hands until it mostly fills the space and is about the same thickness all over. Use the tips of your fingers to tap down the dough all over, leaving little finger-shaped holes over the whole surface. Cover again with a damp towel, putting something over it like a sheet pan or cutting board to keep it in place without pressing on the dough too much. Leave it in a warm place for about 45 minutes.
After the 45 minute rest, preheat your oven to 475° F. The dough will have puffed and bubbled up a bit. Distribute your jalapenos over the surface of the dough, pressing them into the dough a bit with your fingertips as above. They don't need to be all the way in the dough, just a bit pressed down here and there so they don't all fall off the top when you're eating the focaccia. Pour the remaining oil over the surface and sprinkle with salt. Let your dough rest about 20 minutes.
Bake the focaccia in the bottom third of the oven for 40 minutes, turning the pan around once or twice to be sure it bakes evenly. If it starts to get too dark on top, take the whole pan out and quickly cover it in tin foil before putting it back into the oven for the remainder of the cooking time. When it's done, it will be golden all over and darker brown in some places, the edges of the bread will be golden and pulling away from the sides of the pan.
Enjoy your jalapeno focaccia as soon as it's cool enough to eat. Keep leftovers in the fridge or wrap individual slices and freeze for up to 3 months.