In a small bowl, dissolve sugar and yeast in warm water. Let stand until foamy, approximately 10 minutes.
In a large bowl, combine the flour with thyme, salt, and pepper. Add the foamy yeast mixture and stir well to combine. Add in additional water, 1 Tbsp at a time, until the dough forms a ball (be sure to not add too much water as the lentils will carry some moisture). Add the lentils into the dough and knead on a lightly floured surface for 1 minute.
Put the oil in the bowl and roll the dough around to coat (this prevents the dough from drying out). Cover the bowl with a damp cloth, set in a warm place and allow it to double in size, approximately 25-30 minutes. During this time, preheat the grill to 400°F and prepare the toppings.
To prepare the toppings, lightly sauté the onions, tomatoes, and lentils in oil. Sauté just until the onions are rendered. Season with salt and pepper as desired. Cool the mixture slightly and toss in the arugula.
Once the dough has risen, turn it out on a lightly floured surface; knead briefly. Roll into a rectangle shape, 1/2-inch thick. Brush the dough with oil and season with salt. Place the oiled side down on the clean grill, brush with oil and season the other side of the dough (top facing). Cook for 6-8 minutes with the lid down, checking to make sure it’s not getting too dark. Flip the dough over and put the prepared toppings on the grilled crust side. Sprinkle the cheese evenly over the toppings. Cook for 8-10 minutes or until desired doneness. Please note the temperature range of the BBQ and be attentive to the crust.
Remove the cooked focaccia from the grill and allow to rest for 5 minutes before serving.