In a large frying pan, cook the bacon until crisp. Reserve 1 tablespoon of the bacon fat. Allow to cool, then chop into pieces no larger than 1-cm x 1-cm.
In a small bowl, mix the yeast and the milk together. Let sit for 5 minutes.
In a large bowl, mix together the flour, salt and baking powder. Make a well in the middle and add in the eggs, maple syrup and yeast mixture. Mix everything together until a dough forms, about 5 minutes. You can do this by hand or using the dough hook on a stand mixer.
Add in the bacon and cheeses. Continue kneading for about 10 more minutes, until the dough is smooth and pliable. If kneading by hand, pour the dough out onto a silicon mat. If the dough is too wet, add in flour 1 tablespoon at a time. The dough will be sticky, but you should be able to handle it.
Transfer the dough into a large bowl. Cover and let rest until the dough has doubled in size, about 1 to 1 1/2 hours (depending on the temperature of your kitchen).
When the dough is ready, pour it out of the bowl onto a floured surface. Fold the dough twice. Divide the dough into 8 equal pieces (for thicker buns) or 12 equal pieces (for thinner buns). In the photos above, I divided it into 8 equal pieces. Roll each piece into a loose ball and flatten into a disk about 2-cm tall. Set the disks on a floured baking tray and allow to rest for 20 minutes.
Preheat oven to 400°F/205°C. Bake the buns for 25 minutes. Immediately after you remove the buns from the oven, brush on the butter and sprinkle on the poppy seeds.
Make the burger patties
Mix the ground beef with salt and pepper. Divide the meat into three equal pieces. Flatten each piece into a disk about 2-cm thick. Using a round cookie cutter, cut out the middle of each patty. Take the cut outs and form a fourth burger patty. Using a round cookie cutter, cut out the middle of the last patty. You can use the last cut out for a mini slider.
Grill the burger patties
Add oil into a medium frying pan and heat on medium heat. Carefully transfer the burger patties onto the pan. Cook for 2 minutes then flip over. As soon as you flip it over, crack an egg into the hole in the burger. Depending on the size of the hole and the size of your egg, the egg white may spill out onto the top of the burger. It's fine if that happens - simply scrape it off the burger using a spatula as it cooks. Cover the frying pan and cook for an additional 3 minutes.
Carefully transfer the burger patty off the pan, being careful not to break the yolk. Depending on the size of your pan, you may need to cook the burger patties in batches.
Serve the burgers
Slice each bacon brioche bun in half. Layer avocado slices on the bottom half. Top with the burger patty. If desired, spread mayonnaise and ketchup to the top bun. Close the burger.