Learn Hardcore Turkish: Kıymayı tavada kavurup yoğurtlu sos ekliyorum. - I sauté the minced meat in the pan and add a yogurt-based sauce.

kavurmak: to brown/roast/sauté (often with little fat; drive off moisture; common with mince/onions).
kızartmak: to fry (often deep- or shallow-fry until golden).
sotelemek: to sauté quickly over high heat with a little oil (loanword, closer to French “sauté”).

It’s the accusative case marker, showing the direct object is specific/definite: kıymayı = the minced meat. Without it (kıyma) the meaning is indefinite/some minced meat.

Why is there a -y- in kıymayı?

It’s a buffer consonant used when a suffix starting with a vowel follows a word ending in a vowel. kıyma + ı → kıymayı. The same happens in forms like arabayı, elmayı.

Why doesn’t yoğurtlu sos take the accusative too?

Because it’s indefinite here: “I add some yogurt sauce.” If it’s specific/previously known, you’d say yoğurtlu sosu ekliyorum (“I add the yogurt sauce”).

Could both objects be accusative at once?

Yes, if both are definite and each belongs to its own verb: Kıymayı tavada kavurup yoğurtlu sosu ekliyorum = I brown the mince and add the yogurt sauce. Indefinite objects remain unmarked.

What exactly is kavurup?

It’s the converb (linking) form with -ıp/-ip/-up/-üp, meaning “and (then) [do this]”. Only the last verb carries tense/person: kavurup … ekliyorum = “(I) brown … and (then) add.”