There isn’t any historical consensus regarding the origins of leavened bread. However, the earliest archaeological evidence that we have happens to be from ancient Egypt:
The development of leavened bread can also probably be traced to
prehistoric times. Yeast spores occur everywhere, including the
surface of cereal grains, so any dough left to rest will become
naturally leavened. Although leavening is likely of prehistoric
origin, the earliest archaeological evidence is from ancient Egypt.
Scanning electron microscopy has detected yeast cells in some ancient
Egyptian loaves (Wikipedia).
Whether or not Egyptians invented the technology of “thick bread,” the Pentateuch suggests that yeast and leavening were strongly associated with Egyptian culture:
17 “Celebrate the Festival of Unleavened Bread, because it was on this
very day that I brought your divisions out of Egypt. Celebrate this
day as a lasting ordinance for the generations to come. 18 In the
first month you are to eat bread made without yeast, from the evening
of the fourteenth day until the evening of the twenty-first day. 19
For seven days no yeast is to be found in your houses. And anyone,
whether foreigner or native-born, who eats anything with yeast in it
must be cut off from the community of Israel. 20 Eat nothing made with
yeast. Wherever you live, you must eat unleavened bread” (NIV Exodus 12:17-20).
One way of understanding the law to have no yeast in one’s house from the 14th to the 21st of the first month, is that Egypt and Egyptian culture was associated with the unique type of bread that they ate which was more thick and airy than the other types of bread eaten in the Middle East at that time. Therefore, the act of eating only non-leavened bread for the holiday of Passover (“the Festival of Unleavened Bread”) was a gesture of completely rejecting Egyptian culture as a whole.
The Midianite dialog in Judges 7 corroborates this theory: the Israelites took from Egypt the technology and cultural convention of baking large and round loaves of bread and this was distinct from all the surrounding cultures. Therefore, a cake of barley bread which is able to roll would unambiguously evoke the culturally unique Israelite nation and their Egyptian origins.