Learn Turkish: Patronumuz bizi eleştireceğine başarılı projelerimizi takdir etmeli. - Instead of criticizing us, our boss should appreciate our successful projects.

This is the structure you are currently learning! It is built from the verb root eleştir- (criticize) + -ecek (future participle) + -i (3rd person possessive "his/her") + -n- (buffer letter) + -e (dative case). Notice that the k in -ecek softens to ğ because it is followed by a vowel. In Turkish, adding the dative case to a future possessive participle creates the meaning "instead of [someone] doing [verb]".

Why is the 3rd person possessive -i used in eleştireceğine if the sentence starts with patronumuz (our boss)?

Even though it is our boss (marked by the 1st person plural -umuz), the boss themselves is a single person ("he" or "she"). Therefore, the boss is a 3rd person singular subject. The possessive suffix on the converb matches who is doing the action, so we use -i to mean "instead of his/her criticizing".

Why do both biz and projelerimiz take an extra -i at the end (bizi, projelerimizi)?

They are both specific direct objects receiving an action, so they need the accusative case suffix (-i). Bizi (us) is the direct object of the criticizing (eleştireceğine), while projelerimizi (our projects) is the direct object of the appreciation (takdir etmeli).