It's clear that when a stone turns into a loaf of bread it not adding dough to the stone. Mt 4:3 is in the same section of BDAG as J 1:14.
BDAG γινομαι 5. to experience a change in nature and so indicate entry into a new condition, become someth. a. w. nouns (Lamellae Aur. Orphicae ed. AOlivieri 1915, p. 16, 5 θεὸς ἐγένου ἐξ ἀνθρώπου [IV/III]; Arrian, Anab. 5, 26, 5; Sir 51:2; 1 Esdr 4:26; Wsd 8:2; 4 Macc 16:6; En 103:11; Tat. 19, τοῦ θανάτου καταφρονηταὶ γίνεσθε): ὅπως γένησθε υἱοὶ τοῦ πατρὸς ὑμῶν that you may become sons of your father Mt 5:45; ποιήσω ὑμᾶς γενέσθαι ἁλιεῖς ἀνθρώπων I will turn you into fishers of people Mk 1:17; a traitor Lk 6:16; friends 23:12 (cp. Jos., Ant. 11, 121); children of God J 1:12; children of light 12:36; a Christian Ac 26:29; apostle AcPlCor 2:4; a father Ro 4:18; a fool 1 Cor 3:18; a spectacle 4:9; a man, an adult 13:11 (Tob 1:9); a curse Gal 3:13. οὐχ ἑαυτὸν ἐδόξασεν γενηθῆναι ἀρχιερέα he did not exalt himself to be made high priest Hb 5:5; ἐγένετο ἀντὶ αὐτοῦ Σαμουήλ Samuel became (high priest) in his place GJs 10:2. W. double nom. (Ps.-Apollod., Epit. 3, 15 δράκων λίθος ἐγένετο; Quint. Smyrn. 12, 507; Bel 28; 4 Macc 18:7)
οἱ λίθοι ἄρτοι γίνονται the stones turn into loaves Mt 4:3. τὸ αἵμα αὐτοῦ λίθον γεγενημένον GJs 24:3.
ὁ λόγος σὰρξ ἐγένετο J 1:14 (the reverse PBerl 13044, col. III, 28ff [UWilcken, SBBerlAk 1923, 161f] τί ποιῶν ἄν τις γένοιτο θεός;). τὸ ὕδωρ γενήσεται πηγή 4:14. ἡ περιτομὴ ἀκροβυστία γέγονεν Ro 2:25. ἐγενόμην ἐγὼ διάκονος I became a courier Col 1:23 (cp. Herodian 2, 6, 8 ἀνὴρ ἔπαρχος γενόμενος).—Also γ. εἴς τι (Menand., Peric. 49f Kö. [169f S.] τὸ κακὸν εἰς ἀγαθὸν ῥέπει γινόμενον; 1 Km 4:9; Jdth 5:18; 1 Macc 2:11, 43; 3:58; En 19:2 al.; B-D-F §145, 1): ἐγένετο εἰς δένδρον it became a tree Lk 13:19; εἰς κεφαλὴν γωνίας Mt 21:42; Mk 12:10; Lk 20:17; Ac 4:11; 1 Pt 2:7 (all in ref. to Ps 117:22); εἰς χαρὰν γ. change (or, turn) into joy J16:20. εἰς οὐδέν come to nothing Ac 5:36. εἰς παγίδα Ro 11:9 (Ps 68:23); εἰς κενὸν γ. be done in vain 1 Th 3:5. εἰς ἄψινθον Rv 8:11. Cp. AcPl Ha 6, 6. Also w. γίνεσθαι omitted: εἰς κατάκριμα (sc. ἐγένετο τὸ κρίμα) Ro 5:18.