Place a large pot of water with a tight-fitting lid over high heat; bring to a boil; add in salt and the pasta.
Cook according to package directions until al dente; right before draining, reserve about ½ cup pasta cooking liquid.
Add olive oil and sliced garlic to a large skillet; place skillet over medium heat, spread the sliced garlic out in an even layer in the heating oil, and keep an eye on the garlic (it can go from golden to burnt in a couple of seconds.
Have a slotted spoon and some paper towels on a plate nearby; once the garlic is toasty brown, remove it from the oil with a slotted spoon and drain it on the paper towels.
Turn the heat up under the skillet w/ oil to medium-high; add in the red pepper flakes, onion, thyme, salt, and pepper.
Cook, stirring often, for about 5 minutes or until the onions become tender and lightly browned.
Add in the wine and cook 2 minutes, then add the chicken stock; continue cooking 2 minutes.
Add in the peas and a couple of ladles of the pasta cooking liquid.
Bring the sauce back to a bubble; with the back of a spoon, mash about half of the peas in the skillet.
Add in the parsley and toasted garlic slices; stir to combine, then taste it; add more salt/pepper if needed.
Add the cooked drained pasta; toss to coat (if the sauce is too thick, add another glug of chicken stock).
Stir in the cheese, then transfer the pasta to serving plates; pass extra cheese at the table.