If using oven, adjust rack to lowest shelf and preheat to 275°. If not, have an Instant Pot ready. Lightly salt pork cubes and pack them in a large Dutch oven, casserole dish, or optional Instant Pot.
Pour oil over pork and sprinkle in the thyme, onion, garlic, and bay leaf. Cover. Place in oven for 3 hours. If using an Instant Pot, select PRESSURE COOK, set to 15 minutes. Leave on natural release for 10 minute and then open valve to manually release the remaining pressure.
When cooking is complete, remove cover and discard bay leaf, onion and garlic with tongs. Set a large strainer over heatproof bowl and carefully pour pork mixture into it. Reserve liquids.
Transfer pork chunks to the bowl of a stand mixer fitted with a paddle attachment. Begin breaking down the pork on low speed and gradually increase speed until pork is finely shredded. With the mixer on fast, add reserved juices a few tbsp at a time until the mixture is creamy. Salt liberally as saltiness will diminish after chilling.
Spoon mixture into 4 oz. (better fit for charcuterie boards) or larger sealed jars a little bit at a time. Press down to remove any air bubbles. Flatten top, wipe rims, and cover with ¼ inch of oil before securing lids on tight and placing the jars in the refrigerator for at least 2 hours and up-to 2 weeks before serving. Tightly sealed jars of rillette may be frozen for months and defrosted in the refrigerator overnight.