Make the brioche:
In a heatproof bowl, combine the eggs with 1/4 cup water and whisk together.
Set the bowl over a small pot of simmering water and whisk the eggs until they reach 110 degrees F on an instant-read thermometer. Remove from the heat, sprinkle the yeast over the warm eggs, and whisk well. Set aside until the yeast is dissolved, 3 to 4 minutes.
Sift together the bread flour, sugar, and salt. Place in a bowl of an electric mixer fitted with the dough hook attachment. Mix on low speed for 30 seconds.
Pour in the egg/yeast mixture. Increase to medium speed and mix for 5 minutes. The dough will be a bit heavy; don't worry!
Add the butter on low speed, piece by piece. When all the butter is in the bowl, increase the speed to medium and mix for 4 to 5 minutes. The dough will be very soft. Place it in a lightly oiled bowl, cover with plastic, and set in a warm area for 30 minutes.
After the first 30 minutes, punch down the dough and set a timer for 30 minutes. Repeat this for a total of 2 hours, punching down the dough 4 times. Make the topping before the final punch.
Make the topping:
In the bowl of a stand mixer fitted with the paddle attachment, cream the butter, sugar, and espresso on high speed until light, about 3 minutes.
Sift together the flour, salt, and cinnamon and add this to the bowl. Mix for 1 minute.
Add the eggs and vanilla bean paste. Mix well for 1 to 2 minutes. Add the half-and-half and mix just until incorporated. Remove the bowl from the mixer.
Punch down the dough for the final time. Press it into a well-buttered 9- by13-inch baking pan, pressing the dough about 1 inch up the sides of the pan. With the tines of a fork, poke the dough all over.
Evenly distribute the toffee and cherries over the dough. Dump the topping over the toffee and cherries and spread it evenly. Let proof in a warm spot for 20 to 25 minutes.
Preheat the oven to 375 degrees F (or for a convection oven, 350 degrees F). Bake the cake for 30 minutes. The topping will be light brown and crusty. Do not over bake. The cake must rest for 1 hour; the topping needs to set up while it cools.
This cake is best served the same day, or the next morning for breakfast.