What New Testament verse (I think it was St. Paul) says that we must first evangelize those who are already in the Church first before making converts of others outside the Church?
You may be thinking of Paul in his letter to the Romans:
For I am not ashamed of the gospel, for it is the power of God for salvation to everyone who believes, to the Jew first and also to the Greek.
9 There will be tribulation and distress for every human being who does evil, the Jew first and also the Greek, 10 but glory and honor and peace for everyone who does good, the Jew first and also the Greek.
Paul, who considered himself (confirmed by the other apostles) to be Apostle to the Gentiles, might have received the evangelizing priority of the Jews from Jesus himself. This is in line with some gospel passages set before Pentecost, where Jesus and his disciples deliberately focused the announcement of the Kingdom only to the Jews, while making it clear that eventually the Gentiles will be included, as it's obvious from passages in Isaiah.
Relevant passages in Matthew about the priority of the Jews:
Instruction to disciples in Matt 10:5-6:
5 These twelve Jesus sent out, instructing them, “Go nowhere among the Gentiles and enter no town of the Samaritans, 6 but go rather to the lost sheep of the house of Israel.
What Jesus said to the Canaanite woman from the district of Tyre and Sidon in Matt 15:24:
[Jesus] answered, “I was sent only to the lost sheep of the house of Israel.”
Matthew is written after Pentecost, several passages about evangelizing the Gentiles:
16 “Behold, I am sending you out as sheep in the midst of wolves, so be wise as serpents and innocent as doves. 17 Beware of men, for they will deliver you over to courts and flog you in their synagogues, 18 and you will be dragged before governors and kings for my sake, to bear witness before them and the Gentiles.
15 Jesus, aware of this, withdrew from there. And many followed him, and he healed them all 16 and ordered them not to make him known. 17 This was to fulfill what was spoken by the prophet Isaiah:
18 “Behold, my servant whom I have chosen, my beloved with whom my soul is well pleased. I will put my Spirit upon him, and he will proclaim justice to the Gentiles. 19 He will not quarrel or cry aloud, nor will anyone hear his voice in the streets; 20 a bruised reed he will not break, and a smoldering wick he will not quench, until he brings justice to victory; 21 and in his name the Gentiles will hope.”
John Piper wrote a thorough study on Rom 1:16, listing 6 ways on how the Jews have priority over the Gentiles, 3 ways on how the Jews don't have priority, as well as why Paul mentioned the topic in his letter. From his website article To the Jew First, and Also to the Greek, the 6 ways that Jews have priority are:
They are the historic chosen people of God.
They are the guardians of God’s special revelation, the Old Testament Scriptures.
The Messiah and Savior, Jesus, comes to the world as a Jew to Jews.
Salvation is from the Jews, since everyone who is saved is saved by being connected to the covenant with Abraham by faith.
The Jews are to be evangelized first when the gospel penetrates a new region.
The Jews will enter first into final judgment and final blessing.
For more explanation on Jesus's ministry to the House of Israel first, see the gotquestions.org website article Why did Jesus say that He was sent only to the lost sheep of Israel (Matthew 15:24)?