In a medium bowl, combine the yeast, 1 tbsp sugar, and warm water. Set aside to proof for 10 minutes. At the end of the 10 minutes, your mixture will be bubbly and frothy. If you do not see this, your water may have been too hot, or your yeast may need to be replaced.
In an extra large bowl/bucket/pot (I use a 6qt bucket), whisk together the flour, 1/2 cup sugar, and salt. I like to use a danish dough whisk, but a regular whisk or rubber spatula will work just fine. Add in 1 egg. In a small bowl, beat the second egg. Add in half of the beaten egg to the flour mix--just eyeball this step. Reserve the other half for later. Add in the honey and vegetable oil and mix together. Add in the yeast mixture and mix until the dough starts to come together. I do the most I can in the bowl/bucket and then transfer to a floured roll mat or clean surface to start kneading. Do not wash your bowl/bucket yet.
Knead the dough until smooth, a bit tacky, and when you poke it, it immediately springs back. This usually takes about 10 minutes. You may need to knead in a bit more flour if the dough still feels sticky. Transfer the dough back to the bowl/bucket and cover tightly with plastic wrap or a lid and let rise in a warm place for 90 minutes. My kitchen tends to be chilly, so I like to preheat the oven to 200 degrees, turn the oven off, and let the dough rise on the stove. The dough will expand and double in size--if you're concerned about the size of your bowl/bucket, you can divide the dough for this step.
Preheat the oven to 350 degrees. Line a baking sheet with parchment paper and set aside. Lightly flour a surface (mat or counter) and transfer the dough to it. Divide the dough into three equal portions. Set aside two of the portions. Divide the dough into three, four, or six equal parts (depending on what kind of braid you'd like to do). We'll walk through six strand braid. Take each portion of dough and roll it into a snake. I like to do this with two hands together, spreading my fingers out as I go so the snake is uniform thickness throughout (picture typing on a computer, rolling outwards). Lightly flour each snake.
I recommend watching the video reference for a quick guide. Take the six strands and lay them out vertically on the mat. Pinch the tops together and fan out the bottoms–the overall shape will now resemble a pyramid. We're always going to start on the right hand side. Take the strand all the way on the right and weave it to the left, over 2 strands, under 1 strand, over 2 strands, settling now all the way on the left. Take the new strand all the way on the right and repeat this process until the whole loaf is braided. Pinch the loose ends together and tuck them under the loaf. Place the braided loaf on the prepared baking sheet. NOTE: You can fit 2 loaves on a standard baking sheet pan (half sheet size).
I like to repeat this process with the second large dough portion, and turn the final large dough portion into challah rolls. To make rolls, divide the dough into six portions and roll each portion into a snake. Individually, tie each snake in a series of knots, tucking the ends underneath. Proceed with the steps below, and bake for a total of 30 - 35 minutes.
Take the reserved egg and mix it with 1 tsp of water. Use a brush to gently egg wash the challah, making sure to get the nooks and crannies. Sprinkle with salt and add any toppings if using. Bake for 22 minutes, rotate your pan and bake for 22 more minutes. If the challah is golden brown and you think it's done, use a potholder to flip the bread over and lightly tap on the bottom of the loaf. If it sounds hollow, the challah is done. Transfer the challah to a wire rack to cool completely prior to slicing.