Combine the water, yeast, and sugar. Let sit about 5 minutes, until the top is nice and foamy.
Mix the flours and salt. Using a stand mixer with a dough hook (or by hand, for a good arm workout) add the yeast mixture to the flours. Stir until it comes together as a ball. Add extra flour or water as necessary until all flour is visibly incorporated and the ball isn't sticky.
Hand knead or use the stand mixer on a slow/medium speed for five minutes.
Cut the dough in half and put extra in the refrigerator or freezer for another use. Cover remaining dough and let rise for 15 minutes.
Shred the cheeses and chop the garlic while the dough is rising. Feel free to play around with the cheese choices. Add the ½ cup broth to a bowl with the broccoli – a little moisture on the broccoli will soften slightly while baking.
Cut the dough into 8 pieces. For each piece, flatten and stretch into a circle about 4 – 5 inches in diameter. Top each dough circle with a little cheese, garlic, broccoli, and more cheese. Wet your fingers with broth and moisten the outer edges of the dough circle. Bring the edges in and seal well, shaping the roll into a nice circle while you do this.
Cover and let rest for 20 – 30 minutes, the rolls will rise a little more but not much.
Bake at 400° for 10 – 15 minutes. Some cheesy may start oozing out before they're done.
Notes
I brushed the tops of mine with oil before baking the last 3 – 5 minutes to get nice golden brown tops.