PERRYTON, Texas—The oil-and-gas well BP PLC is drilling here in the Texas Panhandle looks ordinary enough from the surface. Yet a mile-and-a-half underground, horizontal pipes shoot off for at least a mile in three directions, like a chicken’s foot.

The idea, part of an experiment by BP executive David Lawler, is to make three wells from one. It also is designed to help turn the London-based energy giant into a shale-oil innovator that can better compete with the entrepreneurial outfits that pioneered the business of hydraulic fracturing, or fracking.

Big oil companies like BP are in need of a jolt. The multibillion-dollar projects they specialize in—giant offshore oil rigs and gas-export projects—are often prohibitively expensive in a world of $45-a-barrel oil. U.S. wells are a tempting option, but major oil companies have yet to prove they can master the techniques pioneered by shale drillers, whose innovations fueled a rebirth in U.S. energy production.

If BP, Exxon Mobil Corp. and others can figure out how to coax enough oil out of fracked wells cheaply enough to make it profitable, it could help them maintain production levels. Failure could make it harder to replace the oil from declining older megaprojects, and leave them further behind on innovations transforming the industry.

Six years after the Deepwater Horizon accident in the Gulf of Mexico caused the worst offshore spill in U.S. history, BP is turning again to America. Mr. Lawler, a former college football linebacker turned engineer, is in charge of the push into shale oil and gas. If the Perryton well succeeds, Mr. Lawler could try the same thing with drilling leases BP has in Oklahoma, Texas and beyond, potentially yielding oil and gas on a large scale.