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The secret to burger buns is a lazy autolyse, a long, vigorous knead, the delayed addition of fat, and pressure.
The quantities here match the base Cube Bread recipe just with half the butter, and makes 5 80g rolls.
"Difficult flour" can be the same AP as you use for the rest, but the flavor benefits from having a little flour with character - wheat, or less processed AP flour, or similar. Rye might be overkill, but you do you.
Flour, sugar, and yeast go into a stand mixer bowl. Add milk and mix on low with a dough hook until the dough pulls away from the sides of the bowl, 2-3 minutes. Cover and autolyse 30 minutes, then back with the dough hook on medium a full 20 minutes. If your mixer gets too hot, a wet towel over the motor will help.
After 20 minutes, add the salt and butter. Knead again on medium until the butter and salt is incorporated and it again pulls clear of the edges of the bowl and no residue remains, another 5 minutes. At this point it should windowpane cleanly. It's okay if it nearly does, but if the dough is still leggy, return to the mixer for another 5 minutes.
Proof 90 minutes, knock back, and proof until doubled, another 30-60 minutes.
Your dough should weigh around 400g; divide it into 5 balls. Hold the ball in the circle created between your thumb and index finger and with your other thumb, push the dough up through the ring, slowly constricting the ring as you do. The goal here is a smooth dome of dough at the top, pulled down into a short tail.
Roll the ball lightly on the counter under your cupped hand to pull the tail up into the ball, pinching any seams on the bottom as you go, then, using a small plate or other flat object, press the ball evenly and firmly into the countertop to create a disk 3"-3.5" wide and 1/4" thick. The key here is EVENLY - it is a mistake to think a rise will even out imperfections in the dough, it will instead exaggerate them.
Repeat with remaining dough balls, arranging the dough circles on a silpatted half baking sheet as you go, 1 each per corner and 1 in the middle, 3" or so apart.
Preheat the oven to 400F. Proof the rolls on top of the warming oven under a second inverted half baking sheet until quite large. The time this takes varies, but the rise will accelerate as it goes, so check it after 45 minutes and every 15 minutes after.
Bake on the bottom rack for 12 minutes. If not dark enough at the 12 minute mark they can go another minute or two, but they will brown fast so stay nearby.